The correct answer is $\boxed{\text{B) 1.5 m/sec}}$.
The coefficient of restitution is a measure of how much energy is lost in an elastic collision. A value of 0.5 indicates that half of the energy is lost, while a value of 1 indicates that no energy is lost.
In this case, the ball is moving with a velocity of 5 m/sec and impinges a fixed plane at an angle of 45°. The direction after impact is equally inclined to the line of impact. This means that the ball will bounce off the plane at an angle of 45°, and its velocity after impact will be equal to its velocity before impact, but in the opposite direction.
The coefficient of restitution is given by the equation $e = \frac{v_f – v_i}{v_i – v_f}$, where $v_f$ is the final velocity, $v_i$ is the initial velocity, and $e$ is the coefficient of restitution.
In this case, we know that $v_i = 5$ m/sec, $v_f = -5$ m/sec, and $e = 0.5$. Substituting these values into the equation, we get $0.5 = \frac{-5 – 5}{5 – (-5)}$. Solving for $v_f$, we get $v_f = 1.5$ m/sec.
Therefore, the velocity of the ball after impact is $\boxed{\text{1.5 m/sec}}$.
Option A is incorrect because the velocity of the ball after impact cannot be 0.5 m/sec. The ball is moving with a velocity of 5 m/sec before impact, and the coefficient of restitution is 0.5. This means that the ball will bounce off the plane at an angle of 45°, and its velocity after impact will be equal to its velocity before impact, but in the opposite direction. The final velocity cannot be less than the initial velocity, so option A is incorrect.
Option B is correct because the velocity of the ball after impact is 1.5 m/sec. The ball is moving with a velocity of 5 m/sec before impact, and the coefficient of restitution is 0.5. This means that the ball will bounce off the plane at an angle of 45°, and its velocity after impact will be equal to its velocity before impact, but in the opposite direction. The final velocity cannot be less than the initial velocity, so option A is incorrect. The final velocity cannot be greater than the initial velocity, so option C is incorrect. Option B is the only option that is consistent with the given information.
Option C is incorrect because the velocity of the ball after impact cannot be 2.5 m/sec. The ball is moving with a velocity of 5 m/sec before impact, and the coefficient of restitution is 0.5. This means that the ball will bounce off the plane at an angle of 45°, and its velocity after impact will be equal to its velocity before impact, but in the opposite direction. The final velocity cannot be greater than the initial velocity, so option C is incorrect.
Option D is incorrect because the velocity of the ball after impact cannot be 3.5 m/sec. The ball is moving with a velocity of 5 m/sec before impact, and the coefficient of restitution is 0.5. This means that the ball will bounce off the plane at an angle of 45°, and its velocity after impact will be equal to its velocity before impact, but in the opposite direction. The final velocity cannot be greater than the initial velocity, so option D is incorrect.