Question

A particle initially at rest starts moving from the reference point x = 0 along x-axis, with a velocity v that varies as v = alpha root x, qhere alpha is constant.
A. How do the velocity and acceleration of the particle vary with time?
B. What is the average velocity of the particle over the first s metres of its path?

Answer 1

A particle initially at rest starts moving from the reference point x = 0 along x-axis, with a velocity v varies as $v=\alpha\sqrt{x}$

we know, velocity is the rate of change of displacement per unit time.

so, v = dx/dt, we can use it above expression.

so, dx/dt = $\alpha\sqrt{x}$

or, $\int{\frac{dx}{\sqrt{x}}}=\alpha\int{dt}$

or,$2\sqrt{x}=\alpha t$

squaring both sides,

$4x=\alpha^2t^2$

or, $x=\frac{1}{4}\alpha^2t^2$

now, differentiating x with respect to time,

so, $\frac{dx}{dt}=v=\frac{1}{2}\alpha^2t$ This is required function of velocity varies with time .

again differentiating with respect to time, we get acceleration d²x/dt² = a

$\frac{d^2x}{dt^2}=a=\frac{1}{2}\alpha^2$

average velocity , <v> = $\frac{\int\limits^s_0{v(x)}\,dx}{\int\limits^s_0{dx}}$

= $\frac{2\alpha s\sqrt{s}}{3s}$

= $\frac{2}{3}\alpha\sqrt{s}$