Question

A small ball of mass $m$ starts at a point A with speed $v_0$ and moves along a frictionless track AB as shown. The track BC has coefficient of friction $\mu$. The ball comes to stop at C after travelling a distance L. Find L.

• A. $\frac{2h}{\mu }+\frac{v_0^2}{2\mu g}$
• B. $\frac{h}{\mu }+\frac{v_0^2}{2\mu g}$
• C. $\frac{h}{2\mu }+\frac{v_0^2}{\mu g}$
• D. $\frac{4h}{\mu }+\frac{v_0^2}{2\mu g}$

Answer 1

The correct option is B $\frac{h}{\mu }+\frac{v_0^2}{2\mu g}$

If we $v_o$ be the initial velocity and $h$ be the initial height,
Applying conservation of energy,
$mgh+\frac{1}{2}m{v_o}^2=\frac{1}{2}m{v}^2$
$v^2={v_o}^2+2gh$
Now, if $a$ be the decelaration produced because of friction,
Using, Newton's third equation of motion,
$v^2=u^2+2as$
$0={v_o}^2+2gh-2aL$
$L=$ $\frac{{v_o}^2+2gh}{2a}$
Now, $f=$$\mu N$
where, $N=mg$
$f=$ $\mu mg$
$a=$ $\mu g$
Putting back this value,
$L=$ $\frac{{v_o}^2+2gh}{2\mu g}$

$=\frac{h}{\mu }+\frac{{v_0}^2}{2\mu g}$