Question

You are driving at 40mph which for your tire size equates to 50 rad/s. You accelerate uniformly to a speed of 48mph in 5 seconds. How many revolutions did your tire make?

Answer 1

Answer on Question #48378, Physics, Mechanics | Kinematics | Dynamics

You are driving at 40mph which for your tire size equates to 50 rad/s. You accelerate uniformly to a speed of 48mph in 5 seconds. How many revolutions did your tire make?

Solution:

Linear speed = radius × angular speed

$$v = r \omega$$

Thus, the radius of tire is

$$r = \frac {v}{\omega} = \frac {(40 * 0.44704) \mathrm {m / s}}{50 \mathrm {rad / s}} = 0.358 \mathrm {m}$$

The acceleration is

$$a = \frac {v - v _ {0}}{t} = \frac {(48 - 40) * 0.44704}{5} = 0.715 \mathrm {m / s ^ {2}}$$

Kinematics equation

$$S = v _ {0} t + \frac {a t ^ {2}}{2}$$

where $a$ is acceleration, $S$ is distance, $v_0$ is initial velocity and $v$ is final velocity.

$$S = 40 * 0.44704 * 5 + \frac {0.715 * 5 ^ {2}}{2} = 98.36 \mathrm {m}$$

The number of revolutions is

$$N = \frac {S}{2 \pi r} = \frac {98.36}{2 \pi * 0.358} = 43.7 \approx 44 \mathrm {rev}$$

Answer: $N = 44$ rev

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