First slide

Instantaneous acceleration

Question

A particle starts from point A moves along a straight line path with an aceleration given by a = p - qx where p,q are constants and x is distance from point A. The particle stops at point B. The maximum velocity of the paticle is

Moderate

A

$\large \frac{p} {q}$
B

$\frac{p} {{\sqrt q }}$
C

$\frac{q} {p}$
D

$\frac{{\sqrt q }} {p}$

Solution

Acceleration a = (p - qx)
The velocity is maximum when dv/dx = 0
Now,
$a = v\frac{{dv}} {{dx}} = (p - qx)$

velocity is zero when a = p - qx = 0 i.e. x = p/q
Now, v. dv = (p - qx)dx
$\Rightarrow \int_{0}^{x_{m}} v d v = p \int_{0}^{\frac{p}{q}} d v - q \int_{0}^{\frac{p}{q}} x d x \Rightarrow v_{m} = \frac{p}{\sqrt{q}}$

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