0.7 1.8 acceleration  (Page 4/8)

The graphs of position, velocity, and acceleration vs. time for the trains in [link] and [link] are displayed in [link] . (We have taken the velocity to remain constant from 20 to 40 s, after which the train decelerates.)

Calculating average velocity: the subway train

What is the average velocity of the train shown 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. ${x\prime }_{\mathrm{f}}=3\text{.75 km}$ , ${x\prime }_0=\text{5.25 km}$ , $\mathrm{\Delta }t=\text{5.00 min}$ .

2. Determine displacement, $\mathrm{\Delta }x\prime$ . We found $\mathrm{\Delta }x\prime$ to be $-\text{1.5 km}$

3. Solve for average velocity.

$\stackrel{-}{v}=\frac{\mathrm{\Delta }x\prime }{\mathrm{\Delta }t}=\frac{-\text{1.50 km}}{\text{5.00 min}}$

4. Convert units.

$\stackrel{-}{v}=\frac{\mathrm{\Delta }x\prime }{\mathrm{\Delta }t}=\left(\frac{-1\text{.}\text{50 km}}{5\text{.}\text{00 min}}\right)\left(\frac{\text{60 min}}{\mathrm{1\; h}}\right)=-\text{18}\text{.0 km/h}$

Discussion

The negative velocity indicates motion to the left.

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. In fact, a negative acceleration will increase a negative velocity. For example, the train moving to the left could speed up by an acceleration to the left. In this 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.

Section summary

• Acceleration is the rate at which velocity changes. In symbols, average acceleration     $\stackrel{-}{a}$ is
  $\stackrel{-}{a}=\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}=\frac{v_{\mathrm{f}}-v_0}{t_{\mathrm{f}}-t_0}\text{.}$
• The SI unit for acceleration is ${\text{m/s}}^2$ .
• Acceleration is a vector, and thus has a both a magnitude and direction.
• Acceleration can be caused by either a change in the magnitude or the direction of the velocity.
• Instantaneous acceleration $a$ is the acceleration at a specific instant in time.
• Deceleration is an acceleration with a direction opposite to that of the velocity.

Conceptual questions

Problems&Exercises