Question

While learning to drive, you are in a $1200kg$ car moving at $20.0m/s$ across a large, vacant, level parking lot. Suddenly you realize you are heading straight toward the brick sidewall of a large supermarket and are in danger of running into it. The pavement can exert a maximum horizontal force of $7000N$ on the car.

(a) Explain why you should expect the force to have a well-defined maximum value.

(b) Suppose you apply the brakes and do not turn the steering wheel. Find the minimum distance you must be from the wall to avoid a collision.

(c) If you do not brake but instead maintain constant speed and turn the steering wheel, what is the minimum distance you must be from the wall to avoid a collision?

(d) Of the two methods in parts (b) and (c), which is better for avoiding a collision? Or should you use both the brakes and the steering wheel, or neither? Explain.

(e) Does the conclusion in part (d) depend on the numerical values given in this problem, or is it true in general? Explain

Answer 1

(a) The only horizontal force on the car is the force of friction, with a maximum value determined by the surface roughness (described by the coefficient of static friction) and the normal force (here equal to the gravitational force on the car).

(b) From Newton’s second law in one dimension,

$\sum F_x=m{a}_x:-f=ma\to a=-\frac{f}{m}=(v^2-v_0^2)/2(x-x_0)$

Solving for the stopping distance gives

$x-x_0=\frac{m(v^2-v_0^2)}{2f}=\frac{\left(1200kg\right)[0^2-(20.0m/s{)}^2]}{2(-7000N)}=34.3m$

(c) Newton’s second law now gives

$f=\frac{m{v}^2}{r}$

or, $r=\frac{m{v}^2}{f}=\frac{\left(1200kg\right)(20.0m/s{)}^2}{7000N}=68.6m$

A top view shows that you can avoid running into the wall by turning through a quarter-circle, if you start at least this far away from the wall.

(d) Braking is better. You should not turn the wheel. If you used any of the available friction force to change the direction of the car, it would be unavailable to slow the car, and the stopping distance would be longer.

(e) The conclusion is true in general. The radius of the curve you can barely make is twice your minimum stopping distance.