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Calculating acceleration and distance
With the formula relating distance, acceleration, and time, you can find any of those values, given the other two. If you have an initial velocity, too, finding distance or acceleration isn’t any harder. In this section, we work through some physics problems to show you how these formulas work.
Finding acceleration
Given distance and time, you can find acceleration. Say you become a drag racer in order to analyze your acceleration down the dragway. After a test race, you know the distance you went — 402 meters, or about 0.25 miles (the magnitude of your displacement) — and you know the time it took — 5.5 seconds. So what was your acceleration as you blasted down the track?
Well, you know how to relate displacement, acceleration, and time (see the preceding section), and that’s what you want — you always work the algebra so that you end up relating all the quantities you know to the one quantity you don’t know. In this case, you have
(Keep in mind that in this case, your initial velocity is 0 — you’re not allowed to take a running start at the drag race!) You can rearrange this equation with a little algebra to solve for acceleration; just divide both sides by t2 and multiply by 2 to get
Great. Plugging in the numbers, you get the following:
Okay, the acceleration is approximately 27 meters per second2. What’s that in more understandable terms? The acceleration due to gravity, g, is — 9.8 meters per second2, so this is about 2.7 g-force — you’d feel yourself pushed back into your seat with a force about 2.7 times your own weight.
Figuring out time and distance
Given a constant acceleration and the change in velocity, you can figure out both time and distance. For instance, imagine you’re a drag racer. Your acceleration is 26.6 meters per second2, and your final speed is 146.3 meters per second. Now find the total distance traveled. Got you, huh? “Not at all,” you say, supremely confident. “Just let me get my calculator.”
You know the acceleration and the final speed, and you want to know the total distance required to get to that speed. This problem looks like a puzzler because the equations in this chapter have involved time up to this point. But if you need the time, you can always solve for it. You know the final speed, vf, and the initial speed, vi (which is zero), and you know the acceleration, a. Because
Now you have the time. You still need the distance, and you can get it this way:
The second term drops out because
In other words, the total distance traveled is 402 meters, or a quarter-mile. Must be a quarter-mile racetrack.
Finding distance with initial velocity
Given initial velocity, time, and acceleration, you can find displacement. Here’s an example: There you are, the Tour de France hero, ready to give a demonstration of your bicycling skills. There will be a time trial of 8.0 seconds. Your initial speed is 6.0 meters/second, and when the whistle blows, you accelerate at 2.0 m/s2 for the 8.0 seconds allowed. At the end of the time trial, how far will you have traveled?
You could use the relation
In this case,
You write the answer to two significant digits — 110 meters — because you know the time only to two significant digits (see Chapter 2 for info on rounding). In other words, you ride to victory in about 110 meters in 8.0 seconds. The crowd roars.
Linking Velocity, Acceleration, and Displacement
Say you want to relate displacement, acceleration, and velocity without having to know the time. Here’s how it works. First, you solve the acceleration formula for the time:
Because displacement is
Substituting for the time, t, you get
After doing the algebra and simplifying, you get
Whew! If you can memorize this one, you can relate velocity, acceleration, and displacement. Put this equation to work — you see it often in physics problems.
Finding acceleration
There you are, getting into your Physics racer as the crowd cheers. It’s time for some hefty acceleration. You get out your clipboard. What acceleration would
