Free Fall

In everyday life, we often observe objects falling towards the ground, such as a ball slipping from your hand or a fruit falling from a tree, all due to the force of gravity, explained by Isaac Newton. When an object falls solely under the influence of gravity, with all other forces such as air resistance ignored, its motion is called free fall. In free fall, the object accelerates towards the Earth at a constant rate due to gravity.

Key Characteristics of Free Fall

• Only gravity acts on the object.
• Air resistance is neglected
• Acceleration is constant and equal to g
• Motion is uniformly accelerated
• The direction of motion is always towards the Earth

Initial Velocity in Free Fall

• If an object is dropped → initial velocity u=0
• If an object is thrown downward/upward → u≠0

Important: Free fall does not always mean u = 0

Acceleration Due to Gravity

It is the acceleration offered by the centre of the Earth. The value of acceleration due to gravity is a Universal constant and the value is equal to 9.8 m/sec². When a body only experiences acceleration due to gravity, that means, it is constantly increasing its velocity with 9.8 m/sec every second starting with a zero velocity.

The force during Free fall is nothing but the force of Gravitation acting between the object and Earth. While Solving for the Force acting between the object and earth, the height at which the object is dropped is neglected since it has no value as compared to the radius of the earth (R = 6378 km).

Force of Gravitation ⇢ F = \frac{GMm}{(h+R)^2}

Since, h<<R

Hence, Height at which the object is dropped can be ignored.

F = \frac{GMm}{R^2}

Force is defined as, F=ma

Here, a=g (acceleration due to gravity).

F = mg= \frac{GMm}{R^2}

 mg= \frac{GMm}{R^2}

\boxed {g= \frac{Gm}{R^2}}

Putting the Value of

• G = Gravitational Constant (=6.67×10^-11 m³kg^-1s^-2)
• m = mass of the Earth (=5.9722×10²⁴ kg)
• R = Radius of the Earth (= 6378 km = 6.38 x 10⁶m)

We get : \boxed{g=9.8 \, \, m/s^2 }

Acceleration due to gravity is the acceleration with which objects fall towards the Earth.

• Value near Earth’s surface: 9.8 m/s²
• Direction: vertically downward (towards Earth’s center)

It means velocity increases by 9.8 m/s every second during free fall.

Equations of Motion in Free Fall

From Newton’s equations of motion (for constant acceleration):

1. Velocity-Time Relation: v=u+gt

2. Displacement-Time Relation: h = ut + \frac{1}{2}gt^2

3. Velocity-Displacement Relation: v^2 = u^2 + 2gh

Special Case (Object Dropped)

If u=0

• v=gt
• h = \frac{1}{2}gt^2
• v^2 = 2gh

Graphical Representation of Free Fall

1. Position–Time Graph

• Curved (parabolic) graph
• Shows accelerated motion

2. Velocity–Time Graph

• Straight line with constant slope
• Slope = g (acceleration)

Real Life Scenario for Visualization

When we are Releasing a ball from a height, h then their x-t , v-t & a-t graphs are as follows:

Solved Problems

Question 1: Two objects are dropped together, the other object is 10 times heavier than the first one, while first object falls in 10 seconds, the other takes 12 seconds to fall. Which of the two will obtain more velocity under free fall?

Solution: Formula for velocity under free fall, 

v = gt

Hence, the velocity is independent of the mass or weight. 

V₁ = 9.8 × 10 = 98 m/sec

V₂ = 9.8 × 12 = 117.6 m/sec

Therefore, The second object will have more velocity.

Question 2: What is the height at which a ball is dropped if it takes 9 seconds to reach the ground. Does the mass of the ball is an important factor in order to obtain the height?

Solution: The formula for height of the ball,

h = 1/2 (gt²)

Since, the formula does not include the mass, hence, the value of mass of the ball is not important in order to find the value of height of the ball.

h = 1/2(9.8× 81)

h = 396.9 meters

Question 3: What was the height of an object when the velocity obtained by the object is 10m/sec?

Solution: The formula of equation of motion used here, 

v² = 2.g.h

10= 2 × 9.8 × h

100 = 19.6h

h = 100 / 19.6

h ≈ 5.1 m

Question 4 : An object is dropped from a height of 20 meters. How long does it take to reach the ground? (Take acceleration due to gravity, g=9.8 m/s2)

Solution: s=1/2(gt²)

Here,
s=20 ms
g=9.8 m/s²

So, 20= 1/2×9.8 ×t²
20=4.9t²

t²=20/4.9 ≈4.08

t= 4.08​≈2.02s

Question 5: A ball is thrown straight up with an initial speed of 15 m/s. How long will it take to reach the highest point?

Solution: At the highest point, the velocity becomes zero. Using the equation: v=u-gt

Here,
v=0
u=15 m/s
g=9.8 m/s²

So,
0=15−9.8t
9.8t=15
t=1.5/9.8≈1.53 s

Unsolved Problems

Question 1: A ball is dropped from rest from a height of 45 m. Find the time taken to reach the ground.
(Take g=9.8 m/s2g = 9.8 \, m/s^2g=9.8m/s2)

Question 2: An object is dropped from a tower and reaches the ground in 4 seconds. Find the height of the tower.

Question 3: A stone is thrown vertically downward with an initial velocity of 10 m/s from a height of 80 m. Find:
(a) Time taken to reach the ground
(b) Final velocity just before hitting the ground

Question 4: A ball is thrown vertically upward with a velocity of 20 m/s. Find:
(a) Maximum height reached
(b) Time taken to reach the highest point

Question 5: Two objects are dropped from the same height, but one is dropped 2 seconds after the other. Find the distance between them after 4 seconds from the release of the first object.