Motion Under Gravity Physics Notes

Under gravity, acceleration is 9.8 m/s² and is denoted by g. When an object is falling freely under gravity, then the above equations would be adjusted as follows:

• v = u + gt
• h = ut + 1/2 gt²
• V²= u² + 2gh

In the above equation, + is replaced by – if the body is thrown upwards.

Maximum Height attained

Let a body be projected vertically upwards with an initial velocity u. As it moves upwards its acceleration is taken as −g. As the body goes up its velocity decreases and finally becomes zero (v = 0) when it reaches maximum height. Now the above equation (3) becomes:

-u²= -2gh

From the above, we can derive that: h=u²/2g.

Time of Ascent (t₁)

The time taken by a body thrown up to reach maximum height is called its time of ascent.  Let t₁ be the time of ascent. At the maximum height its velocity v = 0.  Equation (1) becomes

0 = u − gt₁

t₁ =u/g

Time of descent (t₂)

After reaching the maximum height, the body begins to travel downwards like a freely falling body. The time taken by a freely falling body to reach the ground is called the time of descent (t₂). In this case u = 0 and g is positive. Equation (2) becomes

By Equation (4)

The above discussion makes it clear that time of ascent is equal to the time of descent in the case of bodies moving under gravity.

Time of Flight

The time of flight is the time taken by a body to remain in air and is given by the sum of the time of ascent (t1) and the time of descent (t2).

Velocity of a body dropped from a height

When a body is dropped from a height h its initial velocity u is zero. Let the final velocity on reaching the ground is v.

Equation (3) becomes

v2 = 2gh

At the same time, from Equation (4) we note that

This means that:

Velocity of the body falling from a height h on reaching the ground is equal to the velocity with which it is projected vertically upwards to reach the same height h. Hence the upward velocity at any point in its flight is the same as its downward velocity at that point.

Numerical Example
1. A coin was thrown vertically upwards and it rose to a length of 10 metre. What is the velocity with which the body was thrown upwards?
Answer: In this question: h = 10 m, v = 0, u = ?, g = -9.8 ms-²

Using equations: v² – u² = 2gh

0 – u² = -2 x9.8 x10; u² = 196; u = 14 m/s
2. A coin was thrown vertically upwards and it rose to a length of 10 metre. What was the time taken by the body to reach the highest point?
Answer: From the first question u = 14 m/s, v = 0, t = ?

v= u – gt

0 = 14 – 9.8 x t

t = 1.43 second

Practical Questions

When we drop a coin and a feather simultaneously in a tube fill with air and evacuated tube, which one will reach the bottom first?

When we drop a coin and a feather simultaneously in a tube fill with air and evacuated tube, we get the following observations.

• When the tube has air, coin which is heavier than the feather reaches the bottom of the tube more rapidly while the feather flutters down slowly.
• When there is no air in the tube, coin and the feather to fall together.

From this experiment we understand that air resistance affects the motion of a falling body.  The air resistance on a falling body depends on its shape, size and speed.

Is it possible for the acceleration to be decreasing while the velocity increasing during the same interval of time?

Yes, it’s possible. If the acceleration acts in the direction of motion, it will always cause increment in the velocity. If the acceleration is decreasing but acting in the same direction, the rate, of increase of velocity will decrease. Consequently the velocity will continue to increase slowly. For example, in case of a sphere falling in a viscous liquid, the net acceleration decreases but the velocity increases till the sphere attains its terminal velocity.

A beaker is left out in the rains. Will the rate at which the beaker is filled be altered if a horizontal wind starts to blow?

Answer: No. Beaker will be filled with the same rate because filling of beaker depends on vertical component of the rain.

Two balls of different masses are thrown vertically upwards with the same speed. They pass through the point of projection in their downward motion with the same speed (neglect air resistance). This statement is true or false?

Answer: True. In absence of air resistance a ball will return to the point of projection with the same speed.

Terminal Velocity

When a body falls, it accelerates due to gravity and the retarding force of air resistance increases with speed. This continues till the force of air resistance equals the weight of the object. Now the object no longer accelerates but falls with a constant speed called the terminal velocity. The terminal velocity is about 200 km/hr for a skydiver with an unopened parachute. While falling, the skydivers use a “spread-eagle” position to increase the air resistance and prolong the time of fall. When the parachute is opened, the fall is slowed by the additional resistive force.