Question

a man has two spheres a and b. he gently drops sphere a vertically down and throws the sphere b horizontally at the same time. which of the following statements is correct? a both spheres reach the ground at the same time b sphere a reaches the ground earlier c sphere b reaches the ground earlier d cant be predicted

Explanation:

Step1: Analyze vertical - motion components

The vertical - motion of an object in free - fall near the Earth's surface is described by the equation $h = v_{0y}t+\frac{1}{2}gt^{2}$, where $h$ is the height, $v_{0y}$ is the initial vertical velocity, $t$ is the time, and $g$ is the acceleration due to gravity. For sphere A, it is dropped vertically down, so its initial vertical velocity $v_{0yA}$ is non - zero in the downward direction. For sphere B, it is thrown horizontally, so its initial vertical velocity $v_{0yB}=0$. But the vertical acceleration for both spheres is $g$ (downward) and they start from the same height $h$.

Step2: Consider independence of motions

The horizontal and vertical motions of an object are independent of each other. The time it takes for an object to fall from a height $h$ is determined only by its vertical motion. Using the equation $h = v_{0y}t+\frac{1}{2}gt^{2}$, when considering the vertical motion of both spheres, if we assume they start from the same height $h$, and the acceleration due to gravity $g$ is the same for both, the time $t$ it takes for them to reach the ground is given by the solution of the quadratic equation for the vertical motion. Since the vertical displacement and acceleration are the same for both, and the time of fall depends only on the vertical parameters, the time of fall for both spheres is the same.

Answer:

A. Both spheres reach the ground at the same time