First slide

Instantaneous velocity

Question

A body starts from rest with uniform acceleration a. The acceleration of the body as function of time t is given by the equation a = pt, where p is a constant, then the displacement of the particle in the time interval t = 0 to t = t₁ will be

Difficult

A

$\frac{1}{2} p t_{1}^{3}$
B

$\frac{1}{3} p t_{1}^{2}$
C

$\frac{1}{2} p t_{1}^{2}$
D

$\frac{1}{6} p t_{1}^{3}$

Solution

Integrate twice to convert a-t equation into s-t equation.
$∵$
$\Rightarrow$ $\int_{0}^{v} d v = \int_{0}^{t} a \cdot d t \Rightarrow v = \int_{0}^{t} p t \cdot d t = \frac{p t^{2}}{2}$
$∵$
$\Rightarrow$    $\int_{0}^{s} d s = \int_{0}^{t_{1}} v d t \Rightarrow s = \frac{1}{6} p t_{1}^{3}$

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