Question

Two cannons shoot cannonballs simultaneously as shown in the figure. The mass and speed of the cannonball at ground level is half that of the cannonball at height $ H $ . Each cannonball is in the air for more than two seconds. After 1 second from the launch, acceleration of cannonball at ground is $ {a_1} $ and acceleration of cannonball at height is $ {a_2} $ , choose the correct relationship between $ {a_1} $ and $ {a_2} $ .

(A) $ {a_1} = 4{a_2} $
(B) $ {a_1} = 2{a_2} $
(C) $ {a_1} = {a_2} $
(D) $ {a_2} = 2{a_1} $
(E) $ {a_2} = 4{a_1} $

Answer 1

Hint: If air resistance is neglected, the acceleration of a body is independent of the mass of the body. It is entirely dependent on the acceleration due to gravity on the surface of the planet where the body is undergoing the motion.

Complete step by step answer:
In the diagram, we see a cannon placed at a height above the ground and with its tip parallel to the ground, and another cannon placed on the ground but inclined at a particular angle from the horizontal. These two cannons are said to fire a cannonball simultaneously into the air we are to find the acceleration of the two balls after 1s. Also, the ball on the ground is said to be only half the mass of the ball at the height and the initial velocity is also only half the initial velocity of the ball at a height.
Now, to solve this, we note that there is no information about the air resistance in the air acting on either balls, hence we are expected to neglect the air resistance.
If we do so, then from previous knowledge, we know that irrespective of the mass of the body, all objects in free fall accelerate at the same rate. Hence, for our two balls on the acceleration after anytime are equal to each other
Thus, the correct answer is option C.

Note:
For understanding, if the cannons were on the surface of the earth, the acceleration of them both would be $ 9.8m/{s^2} $ . However, in actuality, the acceleration of two balls changes as the cannonball changes height. Recall that acceleration due to gravity is dependent on the height above the planet, hence the acceleration when the ball is at a height $ H $ and when it is at height say $ \dfrac{H}{2} $ is different. These effects are however often negligible for small heights.