Final answer:
The mass M of the largest stone that can be moved by a flowing river varies as the sixth power of the velocity v.
Step-by-step explanation:
To show that the mass M of the largest stone that can be moved by a flowing river varies as the sixth power of the velocity v, let's use the Principle of Conservation of Energy and consider the force required to lift the stone. The force is equal to the weight of the stone, which is given by the mass multiplied by the acceleration due to gravity.
Using the formula for weight, W = mg, we can write the force as F = mg.
Since the force required to lift the stone depends on the velocity v, density p, and acceleration due to gravity g, we can rewrite the force as F = pvg.
Now, assuming the largest stone that can be moved by the river has a constant density, we can express the mass M as M = pV, where V is the volume of the stone.
Using the formula for the volume of a stone, V = Ah, where A is the cross-sectional area of the stone and h is the height, we can rewrite the mass as M = pAh.
Combining the expressions for the force F and the mass M, we have F = Mv = pAhvg. Rearranging the terms, we get M = (F/v)(1/(pAg)).
Since F/v is a constant, we can write it as k. Therefore, M = k(1/(pAg)).
Now, we know that pAg is the weight of the stone, so we can write it as pAg = W. Substituting this into the equation, we have M = k/W.
Since the weight W is proportional to the velocity v, we can write it as W = cv⁶, where c is a constant.
Substituting this into the equation for M, we have M = k/(cv⁶).
Therefore, we can conclude that the mass M varies as the sixth power of the velocity v.