Analysis:
To find the coefficient of restitution between a sphere and a wall, we need to use the formula:

e = -(v2f - v1f)/(v2i - v1i)

where e is the coefficient of restitution, v1i and v2i are the initial velocities of the sphere and the wall, and v1f and v2f are the final velocities of the sphere and the wall after the collision.

Given:
Mass of the sphere, M = m
Initial velocity of the sphere, v1i = 4i - j
Final velocity of the sphere, v1f = i + 3j
Initial velocity of the wall, v2i = 0 (since the wall is stationary)
Final velocity of the wall, v2f = 0 (since the wall remains stationary)

Calculations:
Using the formula for the coefficient of restitution:

e = -(v2f - v1f)/(v2i - v1i)

Substituting the given values:

e = -(0 - (i + 3j))/(0 - (4i - j))
= -(-i - 3j)/(4i - j)

Simplifying the equation, we get:

e = (i + 3j)/(4i - j)

Explanation:
The coefficient of restitution measures the elasticity of a collision. It is a value between 0 and 1, where 0 represents a completely inelastic collision (no rebound) and 1 represents a perfectly elastic collision (full rebound).

In this scenario, the sphere collides with a wall and rebounds in the opposite direction. The coefficient of restitution tells us how much energy is lost or conserved during the collision.

The formula for the coefficient of restitution takes into account the initial and final velocities of both objects involved in the collision. In this case, since the wall is stationary, its initial and final velocities are both 0.

By substituting the given values into the formula, we can calculate the coefficient of restitution. The negative sign indicates that the direction of the rebound is opposite to the initial direction of the sphere.

In conclusion, the coefficient of restitution between the sphere and the wall is (i + 3j)/(4i - j).

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