Terminal Velocity

Scenario

A golf ball dropped from a tall building travels along a vertical line, experiencing significant air resistance along the journey. Sketch the ball’s v-t, s-t and a-t graphs.

Solution

  • Since this motion involves only downward velocity, the “downward is positive” sign convention is adopted.

3 at

  • At the instant the ball is dropped, the ball experiences only its own weight mg. The acceleration at that instant is thus g.
  • Once the ball is moving, air resistance R kicks in to reduce the acceleration.
  • Since R increases with speed, the acceleration eventually reaches zero when R matches mg.
  • \begin{array}{l}a=\frac{{{F}_{net}}}{m}=\frac{mg-R}{m}\\a=g-\frac{R}{m}\end{array}
  • (Strictly speaking, the acceleration only approaches zero asymptotically as the ball only approaches terminal velocity asymptotically)


3 vt

  • At the instant the ball is dropped, the speed of the ball increases at the rate of 9.81 m s-2.
  • As air resistance kicks in, the speed of the ball increases at decreasing rate, eventually reaching the terminal velocity.
  • (Strictly speaking, the ball only approaches terminal velocity asymptotically)


3 st

  • As the ball speeds up, the s-t graph becomes steeper and steeper.
  • The steepness stops increasing when the ball reaches terminal velocity. The s-t graph becomes a straight line whose gradient is equal to the terminal velocity.

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