Question

A sphere of mass $m$ is moving with a velocity $(4\hat{i}-\hat{j})m/s$ hits a frictionless and rebounds with a velocity $(\hat{i}+3\hat{j})m/s$ . The coefficient of restitution between the sphere and the surface is :

• A. $2/5$
• B. $4/9$
• C. $9/16$
• D. $4/25$

Answer 1

The correct option is C $9/16$

The final momentum is given as,

$m_1{v}_1=m\left(i+3j\right)$

The initial momentum is given as,

$m_2{v}_2=m\left(4i-j\right)$

The change in momentum is given as,

$\Delta mv=m\left(i+3j\right)-m\left(4i-j\right)$

$\Delta mv=m\left(-3i+4j\right)$

The component of the initial velocity along impulse is given as,

$u=\frac{\left(4i-j\right)\left(-3i+4j\right)}{\sqrt{9+16}}$

$=-\frac{16}{5}$

The component of the final velocity along impulse is given as,

$v=\frac{\left(i+3j\right)\left(-3i+4j\right)}{\sqrt{9+16}}$

$=\frac{9}{5}$

The coefficient of restitution between the sphere and the surface is given as,

$e=\frac{\frac{9}{5}}{\frac{16}{5}}$

$=\frac{9}{16}$

Thus, the coefficient of restitution between the sphere and the surface is $\frac{9}{16}$.