Find the number of revolutions the tire makes during this motion, assuming no slipping. Answer in units of rev.

Question

A car accelerates uniformly from rest and reaches a speed of 30.9 m/s in 4.6 s. The diameter of a tire is 54.7 cm.

Answer 1

Let's organize what you know first:

v_i = 0 ("from rest")

v_f = 30.9 m/s

t = 4.6 s

d = 2r = 54.7 cm = 0.547 m

You also know that the acceleration is constant, although you don't know what it is.

And what you don't know:

a = ? (the car's acceleration)

x = ? (how far the car travels)

θ = ? (the angle that the car's wheels rotate through)

Ultimately, you are looking for θ, measured in number of revolutions. To figure this out, we will do the following:

1) Calculate r and x

2) Use those results to find θ in radians

3) Convert θ from radians to revolutions.

1)

r = d/2 = 0.547/2 = 0.2735 m

To find x, you can use v_avg = x/t.

When the acceleration is constant, v_avg is the average of the initial and final velocities:

v_avg = (v_i + v_f)/2 = (30.9+0)/2 = 15.45 m/s.

Then x = v_avgt = (15.45 m/s)(4.6 s) = 7.107 m.

2) If there is no slipping, then θ = x/r = (7.107/0.2735) = 25.99 rad

3) There are 2π radians in 1 revolution. You can use that to convert θ from radians to revolutions.