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## Homework Statement

Sand runs from a hopper at constant rate dm/dt onto a horizontal

conveyor belt driven at constant speed V by a motor.

(a) Find the power needed to drive the belt.

(b) Compare the answer to (a) with the rate of change of kinetic

energy of the sand. Can you account for the difference?

## Homework Equations

[tex]P=F{\cdot}v[/tex]

[tex]F=\dot{m}v+ma[/tex] (Since the sand's horizontal velocity component in our assumed reference frame is 0)

[tex]W=\Delta{T}=\int{Fdx}[/tex] (This is a definite integral but I don't know how to show bounds in latex)

## The Attempt at a Solution

Alright, so part (a) is pretty straightforward. a is 0, just the mass is changing, so [tex]F=\dot{m}v[/tex] and then [tex]P=\dot{m}v^2[/tex]

So, now for part (b), which is the part that is confusing me, [tex]T=\frac{1}{2}mv^2[/tex] so [tex]\dot{T}=\frac{1}{2}\dot{m}v^2+mva[/tex] a is 0, so [tex]\dot{T}=\frac{1}{2}\dot{m}v^2[/tex]

So, now we find that the rate of change of kinetic energy is one half the power, which is what I presume they expected us to find by the wording of the question. But, when I start to think about an explanation, what I see is that the reason these aren't equal is because, mathematically, when we derive the equation [tex]W=\Delta{T}=\int{Fdx}=\frac{1}{2}mv_b^2-\frac{1}{2}mv_a^2[/tex] we assume that F=ma and m is a constant. But, if we rederive W with the correct assumptions for this problem [tex]W=\Delta{T}=\int{Fdx}=\int{(ma+\dot{m}v)dx}=\int{\dot{m}vdx}=mv_b^2-mv_a^2[/tex] we find that from our definition [tex]W=\Delta{T}[/tex] that the actual formula we should be using for this problem is [tex]T=mv^2[/tex] Now, if we use this equation for kinetic energy we derived from the correct assumptions for the problem, now we find that [tex]P=\dot{T}[/tex] So, I feel that the answer to part (b) is that the reason for the difference is that we are actually using the wrong equation for kinetic energy in this problem, and if we correct the formula, the two quantities are equal. Will someone tell me if all of this is correct, because I feel like I'm doing more than they've expected me to in this problem.