Vertical Throw (with Air Resistance)

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.

4 at

  • 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 vi and vf respectively.

4 vt

  • Since the ball loses energy to air resistance continuously, the ball’s final speed vf is lower than its initial speed vi.
  • 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.

4 st

  • 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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