More Subway Train Motion Examples

Example on Calculating Average Velocity: The Subway Train

What is the average velocity of the train in part (b) of the first example from the previous lesson, and shown again below, if it takes 5.00 min to make its trip?

Strategy

Average velocity is displacement divided by time. It will be negative here, since the train moves to the left and has a negative displacement.

Solution

1. Identify the knowns. \(x'_{\text{f}} = 3.75\text{ km},\) \(x_0 = 5.25\text{ km},\) \(\Delta t = 5.00\text{ min}.\)

2. Determine displacement, \(\Delta x'.\) We found \(\Delta x'\) to be \(-1.5\text{ km}\) in the first example from the previous lesson.

3. Solve for average velocity.

\(\bar{v} = \cfrac{\Delta x'}{\Delta t} = \cfrac{-1.50\text{ km}}{5.00\text{ min}}\)

4. Convert units.

\(\bar{v} = \cfrac{\Delta x'}{\Delta t} = \left ( \cfrac{-1.50\text{ km}}{5.00\text{ min}} \right ) \left ( \cfrac{60\text{ min}}{1\text{ h}} \right ) = -18.0\text{ km/h}\)

Discussion

The negative velocity indicates motion to the left.

Example on Calculating Deceleration: The Subway Train

Finally, suppose the train in the figure above slows to a stop from a velocity of 20.0 km/h in 10.0 s. What is its average acceleration?

Strategy

Once again, let’s draw a sketch:

As before, we must find the change in velocity and the change in time to calculate average acceleration.

Solution

1. Identify the knowns. \(v_0 = -20\text{ km/h},\) \(v_{\text{f}} = 0\text{ km/h},\) \(\Delta t = 10.0\text{ s}.\)

2. Calculate \(\Delta v.\) The change in velocity here is actually positive, since

\(\Delta v = v_{\text{f}} - v_0 = 0 - (-20\text{ km/h}) = +20\text{ km/h}.\)

3. Solve for \(\bar{a}\)

\(\bar{a} = \cfrac{\Delta v}{\Delta t} = \cfrac{+20.0\text{ km/h}}{10.0\text{ s}}\)

4. Convert units.

\(\bar{a} = \left ( \cfrac{+20.0\text{ km/h}}{10.0\text{ s}} \right ) \left ( \cfrac{10^3\text{ m}}{1\text{ km}} \right ) \left ( \cfrac{1\text{ h}}{3600\text{ s}} \right ) = +0.556\mathrm{\; m/s^2}\)

Discussion

The plus sign means that acceleration is to the right. This is reasonable because the train initially has a negative velocity (to the left) in this problem and a positive acceleration opposes the motion (and so it is to the right). Again, acceleration is in the same direction as the change in velocity, which is positive here. As in the first example from the previous lesson, this acceleration can be called a deceleration since it is in the direction opposite to the velocity.

Sign and Direction

Perhaps the most important thing to note about these examples is the signs of the answers. In our chosen coordinate system, plus means the quantity is to the right and minus means it is to the left. This is easy to imagine for displacement and velocity. But it is a little less obvious for acceleration.

Most people interpret negative acceleration as the slowing of an object. This was not the case in example, where a positive acceleration slowed a negative velocity. The crucial distinction was that the acceleration was in the opposite direction from the velocity.

In fact, a negative acceleration will increase a negative velocity. For example, the train moving to the left in figure is sped up by an acceleration to the left. In that case, both \(v\) and \(a\) are negative. The plus and minus signs give the directions of the accelerations. If acceleration has the same sign as the velocity, the object is speeding up. If acceleration has the opposite sign as the velocity, the object is slowing down.

Check Your Understanding

An airplane lands on a runway traveling east. Describe its acceleration.

Solution

If we take east to be positive, then the airplane has negative acceleration, as it is accelerating toward the west. It is also decelerating: its acceleration is opposite in direction to its velocity.

Video: PhET Interactive Moving Man Simulation

This video by Wassim Ibrahim explains clearly the relation between Position, Velocity and Acceleration through a nice simulator from PHET that shows the data through graphs to better understand the change in these 3 variables in motion

Learn about position, velocity, and acceleration graphs by downloading the 1.7MB simulator here. Move the little man back and forth with the mouse and plot his motion. Set the position, velocity, or acceleration and let the simulation move the man for you.