The velocity of a tennis ball .....

Number Theory Level 2

The velocity of a tennis ball, v v m / s m/s , after t t seconds is given by v = 8 + 10 t − t 2 v = 8 + 10t - t^2 . Work out when the acceleration is 0, hence find out the maximum velocity in m / s m/s .


The answer is 33.

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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1 solution

Tom Engelsman
Jul 8, 2019

The velocity is expressible as a concave-down parabola:

v ( t ) = − ( t − 5 ) 2 + 33 v(t) = -(t-5)^2 + 33

which has a maximum value of 33 m/s at t = 5 t = 5 sec. This also satisfies (by calculus):

a ( t ) = d v d t = 10 − 2 t = 0 ⇒ t = 5 a(t) = \frac{dv}{dt} = 10 - 2t = 0 \Rightarrow t = 5 AND d 2 v d t 2 = − 2 < 0 \frac{d^{2}v}{dt^2} = -2 < 0 (a maximum).

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