Question

A smooth sphere is moving on a horizontal surface with velocity vector 2i+2j immediately before it hits a vertical wall. The wall is parallel to the j vector and the coefficient of restitution between the sphere and the wall is $e = \dfrac{1}{2}$ . The velocity vector of the sphere after it hits the wall is:
A. $i - j$
B. $- i + 2j$
C. $- i + j$
D. $2i - j$

Answer 1

With the given statement problem we can observe that this is an oblique collision and with the help of oblique concept we can calculate the velocity vector of the sphere after it hits the wall. The line of impact is along the i vector and the j vector value will remain constant.

Complete step by step answer:
As per the given problem there is a smooth sphere moving on a horizontal surface with velocity vector 2i+2j immediately before it hits a vertical wall. The wall is parallel to the j vector and the coefficient of restitution between the sphere and the wall is $e = \dfrac{1}{2}$.

We need to find the velocity of the vector of the sphere after it hits the wall.Here we know the initial velocity of the sphere as 2i+2j which hits the wall with coefficients of restitution $\dfrac{1}{2}$ between the wll and the sphere.It is an example of oblique collision because in oblique collision velocity after collision along the line of impact is e times that of initial velocity and velocity perpendicular to the line of impact remain same because there is no force.

Hence we can write the final velocity as,
$V = {V_x}i\, + {V_y}j$
Velocity along the line of impact is ${V_x}$ because here the wall is along the j vector so the line of impact along the i vector.
Hence,
${V_x} = e{U_x}$
Using the statement given in the problem,
$U = {U_x}i + {U_y}j$
$\Rightarrow U = 2i\, + 2j$
Applying it in ${V_x}$ we get,
${V_x} = \dfrac{1}{2} \times 2$
$\Rightarrow {V_x} = 1$
And,
${V_y} = {U_y}$
Hence we get,
${V_y} = 2$
Hence the velocity of the sphere after hitting the wall is given by,
$V = {V_x}i + {V_y}j$
$\therefore V = - i\, + 2j$

Therefore the correct option is $\left( B \right)$.

Note: Here the i term of the velocity becomes negative after hitting this because the sphere bounces back in the same axis but in the opposite direction. Remember before writing the hitting velocity of the sphere make the i vetor term in the negative axis or else your answer will get incorrect.