Power of conveyor belt required to accelerate a mass

Assume the following scenario:

A conveyor belt is driven at velocity v by a motor. Sand drops vertically onto the conveyor belt at a rate of $$m$$ kg/s. What is the power required to move the belt at a consistent speed?

In this case, the rate of change of the kinetic energy of the sand due to its horizontal speed is $$\frac{1}{2}mv^2$$. Which seems to imply that that is the power that the motor is required to provide in order to move the masses.

However, the answer scheme uses conservation of momentum instead of conservation of kinetic energy to arrive at the answer $$\text{Power}=mv^2$$; precisely double the increase in kinetic energy of the sand. Why is this so? What happens to the other $$\frac{1}{2}mv^2$$ of energy per second provided by the motor? Does this mean that exactly half the power provided by the motor is lost to its surroundings? Why is power calculated through momentum and not kinetic energy in this scenario?

Let us consider another scenario:

A block of mass $$M$$ is driven by a motor along a frictionless surface for 5 seconds (after which it turns off) so as to accelerate it to a velocity of 10 m/s. Assuming constant power, what is the power required to be supplied by the motor so as do enable this to happen?

Do I approach this question in kinetic energy or momentum in mind?

The conservation of momentum approach is the correct one as the "collision" between the conveyor belt and the sand is "inelestic" for the following reason.

Just before hitting the belt the horizontal speed of the sand is zero.

Some time after hitting the belt the sand has the same horizontal speed as the belt.

In between kinetic frictional forces between the sand and the belt do work accelerating the sand in the horizontal direction.

The kinetic frictional forces only exist because there is relative movement in the horizontal direction between the sand and the belt.

That relative movement and the kinetic frictional forces result in mechanical energy being converted into heat.

The generation of heat is the reason for the difference between the momentum method and the energy method of solution.

• And the amount of heat generated is equal to the kinetic energy gained by the sand? – David Toh Oct 8 '19 at 8:17
• The way the sums work out that is exactly what happens and it is independent of the magnitude of the frictional force, eg a larger frictional force results in a smaller sliding distance such that the product of force and distance stays the same. You might have met this factor of $\frac 12$ in the energy stored in a spring and the energy stored in a capacitor in electricity? – Farcher Oct 8 '19 at 9:44