Question

Which of the three kicks in Fig. 3-32 is in the air for the longest time if they all reach the same maximum height h. Ignore the air resistance.

(a) a

(b) b

(c) c

(d) all at the same time

FIGURE 3-32 Misconceptual question 8

Short answer

The time of flight for the three balls is the same.

Step-by-step solution

Step 1. Formula of time of flight

The time of flight can be obtained by examining the value of the vertical component of the velocity and the acceleration due to gravity.

Its value is altered linearly to the value of the vertical component of the velocity.

The expression for the time of flight is given by

$t=\frac{2v_y}{g}$.

Here, $v_y$is the vertical component of the velocity, and $g$is the acceleration due to gravity.

Step 2. Horizontal and vertical components of the velocities

From the figure, the distances traveled by the three balls are different due to different angles of inclinations, but the maximum height reached by the three balls is the same. Hence, the vertical component of the velocities of the three balls is the same.

From the expression for the time of flight, you know that the time of flight is directly proportional to the vertical component of the velocity.

As the vertical component of the velocity is the same for all three balls, the three balls reach the same maximum height and remain in the air for the same time.

Therefore, the time of flight for the three balls is the same.

Hence, option (d) is the correct answer.