Calculating the Slow Acceleration of Trains and Their Wheels

In part (a), we are asked to find $x$, and in (b) we are asked to find $\omega$ and $v$. We are given the number of revolutions $\theta$, the radius of the wheels $r$, and the angular acceleration $\alpha$.

The distance $x$ is very easily found from the relationship between distance and rotation angle:

Solving this equation for $x$ yields

Before using this equation, we must convert the number of revolutions into radians, because we are dealing with a relationship between linear and rotational quantities:

Now we can substitute the known values into $x=\mathrm{r\theta }$ to find the distance the train moved down the track:

We cannot use any equation that incorporates $t$ to find $\omega$, because the equation would have at least two unknown values. The equation ${\omega }^2={{\omega }_0}^2+2\text{αθ}$ will work, because we know the values for all variables except $\omega$:

Taking the square root of this equation and entering the known values gives

We can find the linear velocity of the train, $v$, through its relationship to $\omega$:

The distance traveled is fairly large and the final velocity is fairly slow (just under 32 km/h).