Figuring out acceleration without knowing the time length

Question

A car goes from 50mph to 20mph in an unknown amount of time. All that is known is that one of the car's wheels rotated 110 times during the process and that the wheel rotates at a rate that is uniform with the speed. For example, the rotations per minute at 20mph is 20 and at 50mph would be 50. The decrease in velocity was linear. I need to find the decrease in velocity per second that happened during this process.

Normally, I would try and find the difference between the final velocity and the initial velocity and then divide it by the time length in order to find the decrease/increase in velocity per second. However, I do not know how I could utilize the knowledge about the car's wheel rotation. Can someone point me in the right direction?

Answer 1

You need the radius of the wheels, or some equivalent data.

Answer 2

Hint:

If the wheel makes 50 rotations per minute when the vehicle is going 50mph, how many rotations would it make in an hour? And how far would the car have traveled in that time? Use this information to determine the radius of the wheel.

Answer 3

The relation between angular and linear speeds for a rolling wheel is $ \ \omega \ = \ \frac{v}{R} \ $ . This comes from the relation between arclength and angle, $ \ s \ = \ R \theta \ $ , and division by a time interval (or differentiation, if you've had calculus).

At 20 mph = 8.94 m/sec, the tire completes 20 revolutions per minute, which is $ \ 40 \pi \ $ radians per minute, or 2.09 radians per second. Thus,

$$ R \ = \ \frac{v}{\omega} \ \approx \ \frac{8.94 \ \text{m/sec}}{2.09 \ \text{rad/sec}} \ \approx \ 4.27 \ \text{meters} \ \ . $$

A similar calculation for the information at 50 mph gives the same result (Big Wheels! -- the poser probably didn't work this problem out in advance...).

You are given the number of times each wheel turns during the deceleration, so you have the distance over which the deceleration occurs. A useful kinematic equation for constant accleration comes from substituting the equation for velocity [ $ \ v = v_0 + at \ $] into the equation for position [ $ \ x = x_0 + v_0 t + \frac{1}{2} at^2 \ $ ] , producing $ \ v^2 \ = \ v_0^2 \ + \ 2 \ a \ (x-x_0) \ $ . The starting speed was 50 mph = 22.34 m/sec ; with the distance $ \ (x-x_0) \ $ now known, you can solve for the acceleration in $ \ \frac{\text{m}}{\text{sec}^2} \ $ , which will be negative, since this is a deceleration. [I get a very gentle deceleration of about $ \ -0.07 \ \frac{\text{m}}{\text{sec}^2} \ $ (must be those Big Wheels...) . ]