Question

The tires of a car make 78 revolutions as the car reduces its speed uniformly from 91.0km/h to 63.0km/h . The tires have a diameter of 0.88m .
Part A

What was the angular acceleration of the tires?

Express your answer using two significant figures.

Part B

If the car continues to decelerate at this rate, how much more time is required for it to stop?

Express your answer to two significant figures and include the appropriate units.

Part C

How far does the car go? Find the total distance.

Express your answer to three significant figures and include the appropriate units.

Answer 1

First, find out how fast the tyres are spinning at each of the two speeds given. Converting to standard units, we have that

100 km/h = 27.778 m/s
40 km/h = 11.111 m/s

Find out the circumference of the tire to find the relationship between revolutions per second and velocity.

C = pi*diameter = 3.1416*0.80 m = 2.5133 m

So 1 rotation per second is the same as 2.5133 meters per second.
Thus, 27.778 m/s = 27.778 / 2.5133 rps = 11.052 rps
and, 11.111 m/s = 11.111 / 2.5133 rps = 4.4209 rps

Of course, "rotations per second" is not the proper units for angular velocity - for that we need "radians per second". Since one rotation is the same as 360 degrees, or 2*pi radians, the proper angular velocities are:

11.052 rps * 2 * pi = 69.442 rads per second
4.4209 rps * 2 * pi = 27.777 rads per second

The wheels rotated 75 times during this deceleration, thus the car's wheels rotated through an angular distance of 75*2*pi = 471.24 radians

The familiar formula for linear acceleration can be used here too, replacing "distance" with angular distance, velocity with angular velocity, and so on:

v_final^2 = v_initial^2 + 2ad

Solving for angular acceleration,

a = (v_final^2 - v_initial^2) / 2d = [(27.777^2)-(69.442^2)]/(2*471.24)
a = (771.56-4822.2)/(942.48) = -4.2979 rad/s^2

That's the answer to part (a), although you only need to report 2 significant digits, so a = -4.3 rad/s^2

Part (b) is MUCH simpler. Since it is constant acceleration, we might as well keep working with the angular quantities. We want to know how much time it takes for the angular velocity to go from 27.777 rad/s to 0 rad/s with an acceleration of -4.2979 rad/s^2.

a = delta_v / delta_t

therefore delta_t = delta_v / a = -27.777 / -4.2979 = 6.4629 seconds

Again, only 2 significant digits needed, so 6.5 seconds for the car to come to a complete stop.