A timing system records elapsed time (in s) as a strictly increasing

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

A timing system records elapsed time

\tau

(in s) as a strictly increasing, twice-differentiable function of displacement

s

(in m) along a track:

\tau=f(s)

. Consequently, displacement can be expressed as the inverse function

s=f^{-1}(\tau)

. (a) Derive a formula for the velocity

\displaystyle \frac{ds}{d\tau}

in terms of

f'(s)

evaluated at the appropriate displacement. (b) Show that the acceleration

\displaystyle \frac{d^{2}s}{d\tau^{2}}

can be written in the form

-\dfrac{f''(s)}{[f'(s)]^{3}}

, specifying clearly at which displacement value the derivatives of

f

are to be evaluated. (c) For

s=10\text{ m}

the measurements are

f'(10)=4\text{ s/m}

and

f''(10)=0.30\text{ s/m}^{2}

. Compute the instantaneous velocity and acceleration when the particle is at

s=10\text{ m}

. (d) Interpret the sign of the acceleration you found in part (c) in the context of the particle’s motion along the track.

Related theory

Browse the problem library