D.88 [See also E.43. Run-away truck beds are stretches of in- creasingly soft sa

Question

D.88 [See also E.43. Run-away truck beds are stretches of in- creasingly soft sand found at the bottom of steeply inclined roadways, in which vehi- cles whose brakes have failed may be safely brought to rest. Vo [] 0 In this case, the truck, of mass ll, enters the bed at x = 0, with speed instant. Our model shall attribute the effect of the truck bogging down soft bed to a force of kinetic friction with increasing coefficient, viz., A-, whe a constant with dimensions of length. Once the truck comes to rest, it remains the frictional force vanishes from that time onward. Ascertain the distance (i at which the truck stops. Vo at some initial in the increasingly re L is

Explanation / Answer

In this problem,

Initial speed, U = V0

Final speed, V = 0

Since the frictional force is given to be, muk, the decelaration = - F/M

Decelaration (or negative acceleration, a = -(1/M)*(muk) = -(1/M) * (x/L)

Use the Newton's law, V^2 = U^2 + 2*a*S, we get

0 = V0² - 2*(1/M)*(x/L)*x

Rearrange terms to get x²

x² = (1/2)*(ML)*V0²

Take square root, to get the final answer.

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