Question

Driving down a hill inclined at an angle θ with respect to horizontal, you slam on the brakes to keep from hitting a deer. Your antilock brakes kick in, and you don’t skid. (a) Analyze the forces. (Ignore rolling resistance and air friction.) (b) Find the car’s maximum possible deceleration, a (expressed as a positive number), in terms of g, θ, and the relevant coefficient of friction. √ (c) Explain physically why the car’s mass has no effect on your answer. (d) Discuss the mathematical behavior and physical interpretation of your result for negative values of θ. (e) Do the same for very large positive values of θ

Answer 1

a] Downward force along the slope = mgsin

upward friction force along the slope =

Vertically downward force = mg

Normal force = R = mgcos

b]

and deceleration d = - a

so,

this is the maximum deceleration

c] Since the net force itself is mass dependent, the overall effect of mass gets canceled out.

d] As becomes more negative (more downward steep slope), the deceleration is small (or in other words, larger force is required to halt). This is because cos(-) = cos() whereas sin(-) = - sin.

e] Since is positive, the motion is upwards now and so a larger positive value of would mean that the deceleration will be large.