Scenario

A player throws a ping pong ball high into the air for a serve. The ball travels along a vertical line, and makes contact with the bat only when it returns to the same vertical height at which it leaves the player’s hand. Air resistance has a significant effect on the motion of the ball. Sketch the ball’s *v-t*, *s-t* and *a-t *graphs.

Solution

- (xmtutor)
- Since this motion involves both upward and downward velocity, the more intuitive
**upward is positive** sign convention is adopted.

- On the way up, air resistance acts downward in the same direction as gravitational pull. The net force is stronger than
*mg* so the acceleration is downward and larger than *g*.
- On the way down, air resistance acts upward in opposite direction to the gravitational pull. The net force is weaker than
*mg* so the acceleration is downward but smaller than *g*.
- At the peak, the ball is instantaneously at rest and thus experiences zero air resistance. The net force is exactly
*mg* so the acceleration is downward and equal to *g* at this instant.
- The green area is larger than the blue area since they represent
*v*_{i} and *v*_{f} respectively.

–

- Since the ball loses energy to air resistance continuously, the ball’s final speed
*v*_{f} is lower than its initial speed *v*_{i}.
- The
*v-t* graph is steepest at the start, when we have the strongest downward air resistance to boost the effect of gravity.
- The
*v-t* graph is most flat at the end, when we have the strongest upward air resistance to diminish the effect of gravity.
- The green and blue areas are matching since they represent the distance traveled in the upward and downward journey respectively.

–

- Because of air resistance, the
*s-t* graph is asymmetrical.
- The downward journey takes a longer time than the upward journey because the ball travels at a lower speed on the way down than on the way up.

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