The motion of a particle moving along the -axis is described by the equation

MATHEMATICS · Subjective · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

The motion of a particle moving along the

x

-axis is described by the equation

x(t)=5t + 2t^{2}

, where

x

is in metres and

t

in seconds. (a) Obtain expressions for the velocity

v(t)

and acceleration

a(t)

as functions of time. (b) State, with reasons, the geometrical nature (straight line, parabola, etc.) of the graphs of

v

versus

t

and

a

versus

t

. (c) Find the time at which the velocity becomes

25\,\text{m}\,\text{s}^{-1}

and the displacement of the particle at that instant. (d) Draw rough sketches of the

v

–

t

and

a

–

t

graphs, clearly indicating intercepts and slopes, and explain how their shapes reflect the results obtained in parts (a) and (b).