Physics I For Dummies. Steven Holzner

      So far, so good. Now you can plug this average velocity into the equation and get

      And this becomes

      You can also put in rather than just plain t:

      Congrats! You’ve worked out one of the most important equations you need to know when you work with physics problems relating acceleration, displacement, time, and velocity.

Notice that when you derived this equation, you had an initial velocity of zero. What if you don’t start off at zero velocity, but you still want to relate acceleration, time, and displacement? What if you’re initially going 100 miles per hour? That initial velocity would certainly add to the final distance you go. Because distance equals speed multiplied by time, the equation looks like this (don’t forget that this assumes the acceleration is constant):

      

You also see this written simply as the following (where t stands for , the time over which the acceleration happened):

      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, v_f, and the initial speed, v_i (which is zero), and you know the acceleration, a. Because , you know that

      Now you have the time. You still need the distance, and you can get it this way:

      The second term drops out because , so all you have to do is plug in the numbers:

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 , except you don’t start off from zero speed — you’re already moving, so you should use the following:

      In this case, , , and , so you get the following:

      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