Chapter 3 : Projectiles

Question 1: Define Projectile. Derive an expression for the time of flight, maximum height and horizontal range when it is projected from ground.

Answer:

If a body thrown into air under the gravity alone, it is called a projectile and hence it is called projectile motion. For example: If a stone thrown into air horizontally or vertically, it moves under the gravity alone. When a body is thrown into air, the projectile is under action of following two types of motion:

1) The horizontal motion which is supplied initially.

2) The vertical motion which is caused by gravity. 

Now, Projectile is projected in two ways:

a) Projectile is projected from ground.

b) Projectile is projected from height.

Projectile is projected from ground: 

Consider a projectile is projected from O on the ground. Let the projectile is projected with the velocity µ_° usual making angle with horizontal. When the projectile is projected from ground, it moves up and its vertical velocity goes on decreasing. The vertical velocity becomes maximum when it travels a maximum height. At maximum height, the vertical velocity goes on decreasing in downward directions until the projectile hits the ground. Now the upward motion the initial velocity is:

µ = µ_°Sinθ

and, Final velocity (v)=0

Time taken to maximum height= t

Here, v = u +at 

Or, 0= µ_°Sinθ-gt

Or, gt=µ_°Sinθ

thus, t= µ_°Sinθ/g

1) Time of Flight (T):

The  time of flight of the projectile is the time it spends in air.

For this, Total time of flight (T)=t+t=2t=2*µ_°Sinθ/g

T= 2µ_°Sinθ/g

2) Maximum Height (h_max ):

The maximum height of projectile is the maximum height it arise above the point of projection.

For maximum height,

v=0

Then 

v²=u²+2as

Or, 0 = (µ_° Sinθ)² -2gh_max

Or,2gh_max = µ_°² Sin²θ

i.e. h_{max =} µ_°² Sin²θ/2g

3) Horizontal Range  (R):

The horizontal range of projectile is the horizontal distance traveled by the projectile during its time of flight.

For this, 

Acceleration due to gravity (g)=0

then,

v=d/t

Or, d=vt

Or, R= µ_° Cosθ*2µ_°Sinθ/g

Or, R= µ_°² Sin2θ/g

Note : Maximum Range --->> 

We  have, R= µ_°² Sin2θ/g

Here R depends on angle

For maximum Range,

Sin2θ=1  

Or, Sin2θ=Sin 90

Or, 2θ=90

Or, θ=45

Question 2: Derive an expression for the time of flight, maximum height and horizontal range when it is projected from height.

Answer: 

Consider, a projectile from point O at height h from the ground. Let, the projectile is  projected horizontally after getting velocity. When the projectile is released, its motion is affected by:

a) The horizontal velocity

b) The acceleration due to gravity (g)

1) Time of Flight (T):

Initial velocity (µ)=0

Height traveled =h

Time taken (T)=?

then,

s=ut+1/2gt^2

Or, h=0*t+1/2*g*T^2

Or, h= (g/2)T^2

Or, T =

T= √2h/g

2) Horizontal Range (R):
The horizontal range of a projectile is the horizontal distance traveled by the projectile in time (T).
Now,
R=µ_{° * T
R=}  µ_{° *}√2h/g

Question 3:  What would be the effect on the maximum range doubling the initial velocity?

Solution:

Here, the range is given by:
R= µ_°² Sin2θ/g
If the velocity is doubled (i.e. µ^’ = 2 µ_°) then range becomes,

R^’= (µ^’)² Sin2θ/g 

Or, R^’= (2µ^°)² Sin2θ/g

Or, R^’= 4µ^°2 Sin2θ/g 
Or, R^’= 4R

Question 4: Can a  body said to be at rest and in motion at the same time?

Answer:

Yes, when a bus is moving on a road, passenger sitting inside the bus are at rest with respect to each other, whereas the passengers are in motion with respect to surrounding (i.e. trees, building, e.t.c).

Question 5: Can the direction of the velocity of a body change when its acceleration is constant? Give example.

Answer:

Yes, the direction of velocity of a body can changed when its acceleration is constant. For example: If the body moving on a circular track with a constant speed, the direction of velocity can be changes which is obtained by draw the tangent line on the circle. 

Question 6: Can an object eastward velocity while experiencing westward acceleration? Give examples.

Answer:

Yes, example: on the application of brakes on a moving vehicles, the direction of retardation (i.e. acceleration ) on it is opposite to that of its motion. But, velocity becomes in forward directions.