Physics

The motion of an object is called two dimensional, if two of the three co-ordinates required to specify the position of the object in space change time.

In such a motion, the object moves in a plane. For example, a billiard ball moving over the billiard table, an insect crawling over the floor of a room, earth revolving around the sun etc.

Two special cases of motion in two dimension are

1. Introduction of Projectile Motion

A hunter aims his gun and fires a bullet directly towards a monkey sitting on a distant tree. If the monkey remains in his position he will be safe but at the instant the bullet leaves the barrel of gun, if the monkey drops from the tree, the bullet will hit the monkey because the bullet will not follow the linear path.

If the force acting on a particle is oblique with initial velocity then the motion of particle is called projectile motion.

2. Projectile

A body which is in flight through the atmosphere under the effect of gravity alone and is not being propelled by any fuel is called projectile.

Example: (i) A bomb released from an aeroplane in level flight

(ii) A bullet fired from a gun

(iii) An arrow released from bow

(iv) A lavelin thrown by an athlete

3. Assumptions of Projectile Motion

(1) There is no resistance due to air.

(2) The effect due to curvature of earth is negligible

(3) The effect due to rotation of earth is negligible.

(4) For all points of the trajectory, the acceleration due to gravity ' ' is constant in magnitude and direction.

4. Principle of Physical Independence of Motions

(1) The motion of a projectile is a two-dimensional motion. So, it can be discussed in two parts. Horizontal motion and vertical motion. These two motions take place independent of each other. This is called the principle of physical independence of motions.

(2) The velocity of the particle can be resolved into two mutually perpendicular components. Horizontal component and vertical component.

(3) The horizontal component remains unchanged throughout the flight. The force of gravity continuously affects the vertical component.

(4) The horizontal motion is a uniform motion and the vertical motion is a uniformly accelerated or retarded motion.

5. Types of Projectile Motion

(1) Oblique projectile motion

(2) Horizontal projectile motion

(3) Projectile motion on an inclined plane

6. Oblique Projectile

xis.

In projectile motion, horizontal component of velocity , acceleration and mechanical energy remains constant while, speed, velocity, vertical component of velocity , momentum, kinetic energy and potential energy all changes. Velocity, and KE are maximum at the point of projection while minimum (but not zero) at highest point.

(1) Equation of trajectory: A projectile is thrown with velocity at an angle with the horizontal. The velocity can be resolved into two

component along -axis and component along

For horizontal motion (i)

For vertical motion

From equation (i) and (ii)

This equation shows that the trajectory of projectile is parabolic because it is similar to equation of parabola

Note : Equation of oblique projectile also can be

(2) Displacement of projectile : Let the particle acquires a position P having the coordinates just after time from the instant of projection. The corresponding position vector of the particle at time is

rectangular components. vritten as as shown in the figure.

The horizontal distance covered during time is given as

The vertical velocity of the particle at time is given as

Now the vertical displacement is given as

Putting the values of and from equation (ii) and equation (iv) in equation (i) we obtain the position vector at any time as

Note : The angle of elevation of the highest point of the projectile and the angle of projection are related to each other as

Example : When a man jumps over the hurdle leaving behind its skateboard then vertical component of his velocity is changing, but not the horizontal component which matches with the skateboard velocity.

As a result, the skateboard stays underneath him, allowing him to land on it.

Let be the instantaneous vigilocity of projectile at time , direction of this velocity is along the tangent to the trajectory at point .

Direction of instantaneous velocity

or

(4) Change in velocity : Initial velocity (at projection point)

Final velocity (at highest point)

(i) Change in velocity (Between projection point and highest point)

When body reaches the ground after completing its motion then final velocity

(ii) Change in velocity (Between complete projectile motion)

(5) Change in momentum : Simply by the multiplication of mass in the above expression of velocity (Article-4).

(i) Change in momentum (Between projection point and highest point)

(ii) Change in momentum (For the complete projectile motion)

(6) Angular momentum : Angular momentum of projectile at highest point of trajectory about the point of projection is given by

(7) Time of flight : The total time taken by the projectile to go up and come down to the same level from which it was projected is called time of flight.

For vertical upward motion

Now as time taken to go up is equal to the time taken to come down so

Time of flight

(i) Time of flight can also be expressed as : (where is the vertical component of initial velocity).

(ii) For complementary angles of projection and

(a) Ratio of time of flight

(b) Multiplication of time of flight

(iii) If is the time taken by projectile to rise upto point and is the time taken in falling from point to ground level then time of flight or

and height of the point is given by

by solving

(iv) If and are at the same level on trajectory and the time difference between these two points is , similarly and are also at the same level and the time difference between these two positions is then

Fig :

(8) Horizontal range: It is the horizontal distance travelled by a body during the time of flight.

So by using second equation of motion in -direction

(i) Range of projectile can also be expressed as :

(where and are the horizontal and vertical component of initial velocity)

(ii) If angle of projection is changed from to then range remains unchanged. or

The angle of projection is given by

Note : If then or .

(9) Maximum height : It is the maximum height from the point of projection, a projectile can reach.

So, by using s

So a projectile has same range at angles of projection and

), though time of flight, maximum height and trajectories are different.

These angles and are called complementary angles of

(i) Maximum height can also be expressed as
(where is the vertical component of initial velocity). projection and for complementary angles of projection, ratio of range

(ii) when i.e., )

i.e., for maximum height body should be projected vertically upward.

(iii) For angle of projection and , range will be same and equal to .

and are also the complementary angles.

(iv) Maximum range : For range to be maximum

i.e.

and

ie., a projectile will have maximum range when it is projected at an angle of 45 to the horizontal and the maximum range will be .

When the range is maximum, the height reached by the projectile

i.e., if a person can throw a projectile to a maximum distance ,

The maximum height during the flight to which it will rise is .

(v) Relation between horizontal range and maximum height : and

(vi) If in case of projectile motion range is times the maximum height So it falls back to the point of projection after reaching the maximum height.

(iii) For complementary angles of projection and

Ratio of maximum height

(10) Projectile passing through two different points on same height at time and : If the particle passes two points situated at equal height at and , then

(i) Height

and (ii)

Comparing equation (i) with equation (ii)

Substituting this value in equation (i)

(ii) Time and

ie.

(II) Motion of a projectile as observed from another projectile Suppose two balls and are projected simultaneously from the origin with initial velocities and , at angle and , respectively with the horizontal.

The instantaneous positions of the two balls are given by

Ball

Ball

The position of the ball with respect to ball is given by

Thus motion of a projectile relative to another projectile is a straight line.

(12) Energy of projectile : When a projectile moves upward its kinetic energy decreases, potential energy increases but the total energy always remain constant.

If a body is projected with initial kinetic energy ), with angle of projection with the horizontal then at the highest point of trajectory

(ii) Potential energy

(iii) Total energy = Kinetic energy + Potential energy

This is in accordance with the law of conservation of energy

7. Horizontal Projectile

When a body is projected horizontally from a certain height ' ' vertically above the ground with initial velocity . If friction is considered to be absent, then there is no other horizontal force which can affect the horizontal motion. The horizontal velocity therefore remains constant and so the object covers equal distance in horizontal direction in equal intervals of time.

The horizontal velocity therefore remains constant and so the object covers equal distance in horizontal direction in equal intervals of time.

(1) Trajectory of horizontal projectile : The horizontal displacement is governed by the equation

The vertical displacement is governed by

Fig :

(since initial vertical velocity is zero)

By substituting the value of in equation (ii)

(2) Displacerment of Projectile After time horizontal displacement and vertical displacement .

So, the position vector

(3) Instantaneots velocity : Throughout the motion, the horizontal component of the velocity is .

The vertical component of velocity increases with time and is given by

Where is the angle of instantaneous velocity from the horizontal.

(4) Time of flight : If a body is projected horizontally from a height with velocity and time taken by the body to reach the ground is , then

(for vertical motion)

(5) Horizontal range : Let is the horizontal distance travelled by the body

(6) If projectiles and are projected horizontally with different initial velocity from same height and third particle is dropped from same point then

(i) All three particles will take equal time to reach the ground.

(ii) Their net velocity would be different but all three particle possess same vertical component of velocity.

(iii) The trajectory of projectiles and will be straight line w.r.t. particle .

(7) If various particles thrownignie.19 same initial velocity but in different direction then

(i) They strike the ground with same speed at different times irrespective of their initial direction of velocities.

(ii) Time would be least for particle which was thrown vertically downward

(iii) Time would be maximum for particle which was thrown vertically upward.

8. Projectile Motion on An Inclined Plane

Let a particle be projected up with a speed from an inclined plane which makes an angle with the horizontal and velocity of projection makes an angle with the inclined plane.

We have taken reference -axis in the direction of plane.

Hence the component of initial velocity parallel and perpendicular to the plane are equal to and respectively i.e. and

The component of along the plane is and perpendicular to the plane is as shown in the figure i.e. and .

Therefore the particle decelerates at a rate of as it moves from to .

(1) Time of flight: We know for oblique projectile motion or we can say

Time of flight on an inclined plane

(2) Maximum height : We know for oblique projectile motion or we can say

Maximum height on an inclined plane

(3) Horizontal range : For one dimensional motion

Horizontal range on an inclined plane

By solving

(i) Maximum range occurs when

(ii) The maximum range along the inclined plane when the projectile is thrown upwards is given by

(iii) The maximum range along the inclined plane when the projectile is thrown downwards is given by

9. Circular Motion

Circular motion is another example of motion in two dimensions. To create circular motion in a body it must be given some initial velocity and a force must then act on the body which is always directed at right angles to instantaneous velocity.

Since this force is always at right angles to the displacement therefore no work is done by the force on the particle. Hence, its kinetic energy and thus speed is unaffected.

But due to simultaneous action of the force and the velocity the particle

follows resultant path, which in this case is a circle. Circular motion can be classified into two types - Uniform circular motion and non-uniform circular motion.

10. Variables of Circular Motion

(1) Displacement and distance : When particle moves in a circular path describing an angle during time (as shown in the figure) from the position to the position , we see that the magnitude of the position vector (that is equal to the radius of the circle) remains constant. i.e. and the direction of the position vector changes from time to time.

(i) Displacement : The change of position vector or the displacement of the particle from position to the position is given by referring the figure.

Putting we obtain

Fig :

(ii) Distance : The distanced covered by the particle during the time is given as

length of the (iii) Ratio of distance and displacement :

(2) Angular displacement : The angle turned by a body moving in a circle from some reference line is called angular displacement.

(i) Dimension as arc radius

(ii) Units = Radian or Degree. It is some time also specified in terms of fraction or multiple of revolution

(iii) Revolution

(iv) Angular displacement is a axial vector quantity.

Its direction depends upon the sense of rotation of the object and can be given by Right Hand Rule; which states that if the curvature of the fingers of right hand represents the sense of rotation of the object, then the thumb, held perpendicular to the curvature of the fingers, represents the direction of angular displacement vector.

(v) Relation between linear displacement

Fig : and angular displacement

or

(3) Angular velocity (a) : Angular velocity of an object in circular motion is defined as the time rate of change of its angular displacement.

(i) Angular velocity

(ii) Dimension :

(iii) Units: Radians per second or Degree per second.

(iv) Angular velocity is an axial vector.

Its direction is the same as that of For anticlockwise rotation of the point object on the circular path, the direction of , according to Right hand rule is along the axis of circular path directed upwards. For clockwise rotation of the point object on the circular path, the direction of is along the axis of circular path directed downwards.

(v) Relation between angular velocity and linear velocity

(vi) For uniform circular motion remains constant where as for nonuniform motion varies with respect to time.

INOte : it is important to note that nothing actually moves in the direction of the angular velocity vector . The direction of simply represents that the circular motion is taking place in a plane perpendicular to it.

(4) Change in velocity: We want to know the magnitude and direction of the change in velocity of the particle which is performing uniform circular motion as it moves from to during time as shown in figure. The change in velocity vector is given as

For uniform circular motion

So

The direction of is shown in figure that can be given as

(5) Time period circular motion, the time period is defined as the time taken by the object to complete one revolution its circular path.

(i) Units : second.

(ii) Dimension: [MLT]

(iii) Time period of second's hand of watch second.

(iv) Time period of minute's hand of watch minute

(v) Time period of hour's hand of watch hour

(6) Frequency In circular motion, the frequency is defined as the number of revolutions completed by the object on its circular path in a unit time.

(i) Units: or hertz .

(ii) Dimension :

Note : Relation between time period and frequency If is the frequency of revolution of an object in circular motion, then the object completes revolutions in I second. Therefore, the object will complete one revolution in second.

• Relation between angular velocity, frequency and time period Consider a point object describing a uniform circular motion with frequency and time period . When the object completes one revolution, the angle traced at its axis of circular motion is radians. It means, when time radians. Hence, angular velocity

If two particles are moving on same circle or different coplanar concentric circles in same direction with different uniform angular speeds and respectively, the angular velocity of relative to will be

So the time taken by one to complete one revolution around with respect to the other (i.e., time in which complete one revolution around with respect to the other (i.e., time in which completes one more or less revolution around than )

Special case : If and so ., particles will maintain their position relative to each other. This is what actually happens in case of geostationary satellite

(7) Angular acceleration : Angular acceleration of an object in circular motion is defined as the time rate of change of its angular velocity.

(i) If be the change in angular velocity of the object in time interval , while moving on a circular path, then angular acceleration of the object will be

(ii) Units :

(iii) Dimension :

(iv) Relation between linear acceleration and angular acceleration

(v) For uniform circular motion since is constant so

(vi) For non-uniform circular motion

11. Centripetal Acceleration

(1) Acceleration acting on the object undergoing uniform circular motion is called centripetal acceleration.

(2) It always acts on the object along the radius towards the centre of the circular path.

(3) Magnitude of centripetal acceleration,

Fig :

(4) Direction of centripetal acceleration : It is always the same as that of . When decreases, also decreases. Due to which becomes more and more perpendicular to . When becomes perpendicular to the velocity vector. As the velocity vector of the particle at an instant acts along the tangent to the circular path, therefore and hence the centripetal acceleration vector acts along the radius of the circular path at that point and is directed towards the centre of the circular path.

12. Centripetal force

According to Newton's first law of motion, whenever a body moves in a straight line with uniform velocity, no force is required to maintain this velocity. But when a body moves along a circular path with uniform speed, its direction changes continuously i.e. velocity keeps on changing on account of a change in direction. According to Newton's second law of motion, a change in the direction of motion of the body can take place only if some external force acts on the body.

Due to inertia, at every point of the circular path; the body tends to move along the tangent to the circular path at that point (in figure). Since every body has directional inertia, a velocity cannot change by itself and as such we have to apply a force. But this force should be such that it changes the direction of velocity and not its magnitude. This is possible only if the force acts perpendicular to the direction of velocity. Because the velocity is along the tangent, this

force must be along the radius (because the

Fig : radius of a circle at any point is perpendicular to the tangent at that point). Further, as this force is to move the body in a circular path, it must acts towards the centre. This centre-seeking force is called the centripetal force.

Hence, centripetal force is that force which is required to move a body in a circular path with uniform speed. The force acts on the body along the radius and towards centre.

Formulae for centripetal force :

Table 3.1 : Centripetal force in different situation

13. Centrifugal Force

It is an imaginary force due to incorporated effects of inertia. When a body is rotating in a circular path and the centripetal force vanishes, the body would leave the circular path. To an observer who is not sharing the motion along the circular path, the body appears to fly off tangentially at the point of release. To another observer , who is sharing the motion along the circular path (i.e., the observer is also rotating with the body with the same velocity), the body appears to be stationary before it is released. When the body is released, it appears to , as if it has been thrown off along the radius away from the centre by some force. In reality no force is actually seen to act on the body. In absence of any real force the body tends to continue its motion in a straight line due to its inertia. The observer easily relates this events to be due to inertia but since the inertia of both the observer and the body is same, the observer can not relate the above happening to inertia. When the centripetal force ceases to act on the body, the body leaves its circular path and continues to move in its straight- line motion but to observer it appears that a real force has actually acted on the body and is responsible for throwing the body radially out-wards. This imaginary force is given a name to explain the effects of inertia to the observer who is sharing the circular motion of the body. This inertial force is called centrifugal force. Thus centrifugal force is a fictitious force which has significance only in a rotating frame of reference.

14. Work Done by Centripetal Force

The work done by centripetal force is

always zero as it is perpendicular to velocity and hence instantaneous displacement.

Work done Increment in kinetic energy of revolving body Work done

Also

Fig :

Example: (i) When an electron revolves around the nucleus in hydrogen atom in a particular orbit, it neither absorb nor emit any energy means its energy remains constant.

(ii) When a satellite established once in a orbit around the earth and it starts revolving with particular speed, then no fuel is required for its circular motion.

15. Skidding of Vehicle on A Level Road

When a vehicle takes a turn on a circular path it requires centripetal force.

If friction provides this centripetal force then vehicle can move in circular path safely if

Friction force Required centripetal force

This is the maximum speed by which vehicle can take a turn on a circular path of radius where coefficient of friction between the road and tyre is .

16. Skidding of Object on A Rotating Platform

On a rotating platform, to avoid the skidding of an object (mass ) placed at a distance from axis of rotation, the centripetal force should be provided by force of friction.

Centripetal force Force of friction

Hence maximum angular velocity of rotation of the platform is , so that object will not skid on it.

17. Bending of A Cyclist

A cyclist provides himself the necessary centripetal force by leaning inward on a horizontal track, while going round a curve. Consider a cyclist of weight taking a turn of radius with velocity . In order to provide the necessary centripetal force, the cyclist leans through angle inwards as shown in figure.

The cyclist is under the action of the following forces :

The weight acting vertically downward at the centre of gravity of cycle and the cyclist.

The reaction of the ground on cyclist. It will act along a linemaking angle with the vertical.

The vertical component of the normal reaction will balance the weight of the cyclist, while the horizontal component will provide the necessary centripetal force to the cyclist.

and

Dividing equation (i) by (ii), we have

or

(iii)

Therefore, the cyclist should bend through an angle

It follows that the angle through which cyclist should bend will be greater, if

(i) The radius of the curve is small i.e. the curve is sharper

(ii) The velocity of the cyclist is large.

Note : For the same reasons, an ice skater or an aeroplane has to bend inwards, while taking a turn.

18. Banking of A Road

For getting a centripetal force, cyclist bend towards the centre of circular path but it is not possible in case of four wheelers.

Therefore, outer bed of the road is raised so that a vehicle moving on it gets automatically inclined towards the centre.

In the figure (A) shown reaction is resolved into two components, the component balances weight of vehicle

(i)

and the horizontal component provides necessary centripetal force as it is directed towards centre of desired circle

Thus

Dividing (ii) by (i), we have

or

(iv) As

If width of the road, height of the outer edge from the ground level then from the figure (B)

since is very small

From equation (iii), (iv) and (v)

INOte : If friction is also present between the tyres and road then

Maximum safe speed on a banked frictional road

19. Overturning of Vehicle

When a car moves in a circular path with speed more than a certain maximum speed then it overturns even if friction is sufficient to avoid skidding and its inner wheel leaves the ground first

Weight of the car

Speed of the car

Fig :

Radius of the circular path

Distance between the centre of wheels of the car

Height of the centre of gravity of the car from the road level

Reaction on the inner wheel of the car by the ground

Reaction on the outer wheel of the car by the ground

When a car move in a circular path, horizontal friction force

provides the required centripetal force i.e.,

For rotational equilibrium, by taking the moment of forces and about

As there is no vertical motion so (iii)

By solving (i), (ii) and (iii)

It is clear from equation (iv) that if increases value of decreases and for

i.e. the maximum speed of a car without overturning on a flat road is given by

20. Motion of Charged Particle In Magnetic Field

When a charged particle having mass , charge enters perpendicularly in a magnetic field with velocity then it describes a circular path.

Because magnetic force works in the perpendicular direction of and it provides required centripetal force

Magnetic force = Centripetal force

radius of the circular path

Fig:

21. Reaction of Road On Car

(1) When car moves on a concave bridge then

Centripetal force

and reaction

(2) When car moves on a convex bridge

Convex bridge

Centripetal force Fig

and reaction

22. Non-Uniform Circular Motion

If the speed of the particle in a horizontal circular motion changes with respect to time, then its motion is said to be non-uniform circular motion.

Consider a particle describing a circular path of radius with centre at Let at an instant the particle be at and be its linear velocity and be its angular velocity.

Then,

..(i)

Differentiating both sides of w.r.t. time we have

Here, (Resultant acceleration)

(Angular acceleration)

Thus the resultant acceleration of the particle at has two component accelerations

23. (1) Tangential acceleration:

It acts along the tangent to the circular path at in the plane of circular path.

According to right hand rule since and are perpendicular to each other, therefore, the magnitude of tangential acceleration is given by

(2) Centripetal (Radial) acceleration :

It is also called centripetal acceleration of the particle at It acts along the radius of the particle at .

According to right hand rule since and are perpendicular to each other, therefore, the magnitude of centripetal acceleration is given by

Table 3.2 : Tangential and centripetal acceleration

Centripetal acceleration	Tangential acceleration	Net acceleration	Type of motion
			Uniform

			translatory motion
		Accelerated translatory motion
			Uniform circular motion
			Non-uniform circular motion

Note : Here a governs the magnitude of while its direction of motion.

(3) Force : In non-uniform circular motion the particle simultaneously possesses two forces

Tangential force :

Not : In non-uniform circular motion work done by centripetal force will be zero since

In non uniform circular motion work done by tangential force will not be zero since

Rate of work done by net force in non-uniform circular motion rate of work done by tangential force

24. Equations of Circular Motion

For accelerated motion	For retarded motion

Where

25. Motion in vertical circle

This is an example of non-uniform circular motion. In this motion body is under the influence of gravity of earth. When body moves from lowest point to highest point. Its speed decrease and becomes minimum at highest point. Total mechanical energy of the body remains conserved and converts into and vice versa.

(1) Velocity at any point on vertical loop: If is the initial velocity imparted to body at lowest point then velocity of body at height is given by

where is the length of the string

(2) Tension at any point orflolencel loop: Tension at general point , According to Newton's second law of motion.

Net force towards centre = centripetal force

or

As

Fig : 3.41

Table 3.3 : Velocity and tension in a vertical loop

Position	Angle	Velocity	Tension

It is clear from the table that:

,

and

Table 3.4: Various conditions for vertical motion

Velocity at lowest	Condition
point	,	Tension in the string will not be zero at any of the point and body will continue the circular motion.
	Tension at highest point will be zero and body will just complete the circle.
,	Particle will not follow circular motion. Tension in string become zero somewhere between points and whereas velocity remain positive. Particle leaves circular path and follow parabolic trajectory.

	Both velocity and tension in the string becomes zero at and particle will oscillate along semi-circular path.
	velocity of particle becomes zero between and but tension will not be zero and the particle will oscillate about the point

	Both velocity and tension in the string becomes zero at and particle will oscillate along semi-circular path.
	velocity of particle becomes zero between and but tension will not be zero and the particle will oscillate about the point

same throughout the path but the . of the body moving in vertical circle is different at different places.

If body of mass is tied to a string of length / and is projected with a horizontal velocity then : Height at which the velocity vanishes is

Table : Different variables in vertical loop

(4) Motion of a block on frictionless hemisphere : A small block of mass slides down from the top of a frictionless hemisphere of radius . The component of the force of gravity ( ) provides required centripetal force but at point it's circular motion ceases and the block lose contact with the surface of the sphere.

For point , by equating the forcres

For point and , by law of conservation of energy

Total energy at point Total energy at point

K.E. + P.E. = K.E. + P.E.

and from the given figure

By substituting the value of and from eq (ii) and (iii) in eq (i)

Height at which the tension vanishes is

(3) Critical condition for vertical looping : If the tension at is zero, then body will just complete revolution in the vertical circle. This state of body is known as critical state. The speed of body in critical state is called as critical speed.

From the above table

It means to complete the vertical circle the body must be projected with minimum velocity of at the lowest point.

i.e. the block lose contact at the height of from the ground. and angle from the vertical can be given by .

26. Conical Pendulum

This is the example of uniform circular motion in horizontal plane. A bob of mass attached to a light and in-extensible string rotates in a horizontal circle of radius with constant angular speed about the vertical. The string makes angle with vertical and appears tracing the surface of a cone. So this arrangement is called conical pendulum.

The force acting on the bob are tension and weight of the bob.

From the figure

and

Fig : (1) Tension in the string:

(2) Angle of string from the vertical:

(3) Linear velocity of the bob:

(4) Angular velocity of the bob:

(5) Time period of revolution :

Fin ID

ש Consider a projectile of mass thrown with velocity making angle with the horizontal. It is projected from the point and returns to the ground at . Also is the highest point attained by it. (See figure).

(i) In going from to , following changes take place -

(a) Change in velocity

(b) Change in speed

(c) Change in momentum

(d) Change (loss) in kinetic energy

(e) Change (gain) in potential energy (f) Change in the direction of motion

(ii) On return to the ground, that is in going from to , the following changes take place

(a) Change in speed zero

(b) Change in velocity

(c) Change in momentum

(d) Change in kinetic energy = zero

(e) Change in potential energy zero

(f) Change in the direction of motion

o (i) At highest point, the horizontal component of velocity is and vertical component of velocity is zero.

(ii) At highest point, linear momentum of a particle

(iii) Kinetic energy of the particle at the highest point

.

of At highest point, acceleration due to gravity acting vertically downward makes an angle of with the horizontal component of the velocity of the projectile.

o At the highest point, momentum of the projectile thrown at an angle with horizontal is and K.E. K.E. .

os projectile motion, horizontal component of velocity remains constant throughout, whereas vertical component changes and becomes zero at the highest point.

The trajectory of a projectile is parabolic.

or For a projectile, time of flight and maximum height depend on the vertical component of the velocity of projection.

os The range of the projectile is maximum for the angle of projection .

of The maximum range of the projectile is:

(6 When the range is maximum, the height attained by the projectile is :

When the range of the projectile is maximum, the time of flight is :

ES The height attained by a projectile is maximum, when .

It is twice that of height attained, when the range is maximum. ES The time of flight of the projectile is also largest for .

o The trajectory of the projectile is a symmetric parabola only when is constant through out the motion and is not equal to or .

ES If velocity of projection is made times, the maximum height attained and the range become times and the time of flight becomes times the initial value.

of If the force acting on a particle is always perpendicular to the velocity of the particle, then the path of the particle is a circle. The centripetal force is always perpendicular to the velocity of the particle.

If circular motion of the object is uniform, the object will possess only centripetal acceleration.

(5 If circular motion of the object is non-uniform, the object will possess both centripetal and transverse acceleration.

of When the particle moves along the circular path with constant speed, the angular velocity is also constant. But linear velocity, momentum as well as centripetal acceleration change in direction, although their magnitude remains unchanged.

of For circular motion of rigid bodies with uniform speed, the angular speed is same for all particles, but linear speed varies directly as the radius of the circular path described by the particle .

os When a body rotates, all its particles describe circular paths about a line, called axis of rotation.

ש The centre of the circle describe by the different particles of the rotating body lie on the axis of rotation.

(6 Centripetal force where mass of the body.

o Centripetal force is always directed towards the centre of the circular path.

6 When a body rotates with uniform velocity, its different particles have centripetal acceleration directly proportional to the radius .

os There can be no circular motion without centripetal force.

厄 Centripetal force can be mechanical, electrical or magnetic force.

es Planets go round the earth in circular orbits due to the centripetal force provided by gravitational force of the sun.

os Gravitational pull of earth provides centripetal force for the orbital motion of the moon and artificial satellites.

ש Centripetal force cannot change the kinetic energy of the body.

of In uniform circular motion the magnitude of the centripetal acceleration remains constant whereas its direction changes continuously but always directed towards the centre.

os A pseudo force, that is equal and opposite to the centripetal force is called centrifugal force.

ש The and are directed along the axis of the circular path. Their sense of direction is given by the right hand fist rule as follows : If we catch axis of rotation in right hand fist such that the fingers point in the direction of rotation, then the outstretched thumb gives the direction of and

o and are called pseudo vectors or axial vectors.

es For circular motion we have -

(v) are perpendicular to

(vi) and lie in the same plane