Mathematics

Problem

The velocity of a body moving along the

x

-axis is described by the quadratic function

v(t)=ut-\tfrac12 bt^2+c

, where

u=6\,\text{m\,s}^{-2}

b=2\,\text{m\,s}^{-3}

and

c=4\,\text{m\,s}^{-1}

; the motion is considered for

t\ge0

. a. (i) Derive an expression for the instantaneous acceleration as a function of time. b. (ii) Show that the acceleration becomes zero at a time

t_0

and obtain

t_0

in terms of

u

and

b

. Then evaluate

t_0

for the given numerical values. c. (iii) Determine the velocity and the displacement from

t=0

t=t_0

. d. (iv) Using the sign of acceleration and velocity, identify the intervals of time during which the body is speeding up and those during which it is slowing down.

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