Question

An object takes 7 seconds to reach the ground from a height of 49 m. Find the value of g on the planet.

Answer 1

To answer this question, the concept of free-fall under gravity must be understood. Then, the equation of motion relating the displacement, time and the acceleration must be applied to the free-fall to solve for the acceleration.

Complete step-by-step answer:
Free-fall is defined as the motion of the body assisted by gravity alone and no other force acting on it. By applying the Newton’s second law of motion, we have –
F = ma
Here, we see that the force applied on the freely falling body, results in a constant acceleration on the body.
This acceleration is called acceleration due to gravity (g). The value of this acceleration on the surface of Earth is $9 \cdot 81m{s^{ - 2}}$.
Consider an object taking time to execute free-fall from a height s.
The equation of motion relating the displacement, acceleration and time is –
$s = ut + \dfrac{1}{2}a{t^2}$
The value of acceleration during the free-fall is constant throughout the duration of the free-fall. Since the acceleration is constant, the initial velocity of the object can be considered as zero.
Hence, $u = 0$ and $a = g$.
Applying this in the above equation, we get –
$s = \dfrac{1}{2}g{t^2}$
Given,
Height, s = 49 m
Time, t = 7 sec
From the above equation,
$g = \dfrac{{2s}}{{{t^2}}}$
Substituting in the above equation,
$g = \dfrac{{2 \times 49}}{{{7^2}}} = \dfrac{{2 \times 49}}{{49}} = 2m{s^{ - 2}}$
Hence, the acceleration due to gravity in this new planet, $g = 2m{s^{ - 2}}$

Note: Some of the examples of free-fall are: i) A spacecraft crashing down towards the Earth ii) An object thrown out of a drop-tube iii) A projectile thrown very high up in the air.
The students should note that the following objects are not in free-fall: i) Descent of aircraft: There is additional lift acting on the aircraft ii) Descent with the help of parachute: There is a drag force acting in the opposite direction of the free-fall, thereby slowing the descent.