An Accelerating Subway Train

Start by making a simple sketch.

Figure 3.4

This problem involves four steps:

1. Convert to units of meters and seconds.
2. Determine the change in velocity.
3. Determine the change in time.
4. Use these values to calculate the average acceleration.

1. Identify the knowns. Be sure to read the problem for given information, which may not look like numbers. When the problem states that the train starts from rest, you can write down that the initial velocity is 0 m/s. Therefore, v₀ = 0; v_f = 30.0 km/h; and Δt = 20.0 s.

2. Convert the units.

   $$\frac{\text{30}\text{.0 km}}{\text{h}}\text{ × }\frac{{\text{10}}^{\text{3}}\text{m}}{\text{1 km}}\text{ × }\frac{\text{1 h}}{\text{3600 s}}\text{ = 8}\text{.333 }\frac{\text{m}}{\text{s}}$$

   3.1
3. Calculate change in velocity, $\Delta v=v_{\text{f}}\text{– }{v}_{\text{0}}\text{= 8}\text{.333 m/s – 0 = + 8}\text{.333 m/s,}$ where the plus sign means the change in velocity is to the right.

4. We know Δt, so all we have to do is insert the known values into the formula for average acceleration.

   $$\overline{a}=\frac{\Delta v}{\Delta t}=\frac{8.333\text{m/s}}{20.00\text{s}}=+0.417\frac{\text{m}}{{\text{s}}^{\text{2}}}$$

   3.2