Q26DQ - University Physics with Modern Physics

Question

A ball is dropped from rest and feels air resistance as it falls. Which of the graphs in Fig. Q5.26 best represents its vertical velocity component as a function of time?

      

Step-by-Step Solution

Answer

Graph (a) best represents the acceleration of a falling ball as a function of time.

Step 1: Determine the expression for the vertical velocity of the component of the ball as a function of time:

The vertical velocity component of the dropping ball varies with time because of air resistance on the ball and the effect of gravity after the terminal velocity of the ball.

The velocity of the falling ball is given by the following equation:

$\mathrm{v}={\mathrm{v}}_{\mathrm{t}}\left(1-{\mathrm{e}}^{-\left(\mathrm{k}/\mathrm{m}\right)\mathrm{t}}\right)$ 

Here, ${\mathrm{v}}_{\mathrm{t}}$ is the terminal speed of the ball, k is drag coefficient, m is the mass of the falling ball, t is the time for falling.

Step 2: Explanation for correct graph:

The expression for the velocity of dropping a ball is varying exponentially with time. The velocity is zero initially but started increasing with time but becomes after a certain time. The velocity of the ball after this certain time becomes equal to the terminal velocity of the ball.

The acceleration of the falling ball is decreasing exponentially with time so the correct graph for the falling ball will be a graph (a).