Physics I For Dummies. Steven Holzner

easily enough by dividing velocity by time to get acceleration:

In terms of units, the equation looks like this:

      Distance per time squared? Don’t let that throw you. You end up with time squared in the denominator because you divide velocity by time. In other words, acceleration is the rate at which your velocity changes, because rates have time in the denominator. For acceleration, you see units of meters per second², centimeters per second², miles per second², feet per second², or even kilometers per hour².

      

It may be easier, for a given problem, to use units such as mph/s (miles per hour per second). This would be useful if the velocity in question had a magnitude of something like several miles per hour that changed typically over a number of seconds.

      Looking at positive and negative acceleration

      Just as for displacement and velocity, acceleration can be positive or negative. This section explains how positive and negative acceleration relate to changes in speed and direction.

      Changing speed

      The sign of the acceleration tells you whether you’re speeding up or slowing down (depending on which direction you’re traveling).

      For example, say that you’re driving at 75 miles per hour, and you see those flashing red lights in the rearview mirror. You pull over, taking 20 seconds to come to a stop. The officer appears by your window and says, “You were going 75 miles per hour in a 30-mile-per-hour zone.” What can you say in reply?

      You can calculate your rate of acceleration as you pulled over, which, no doubt, would impress the officer — look at you and your law-abiding tendencies! You whip out your calculator and begin entering your data. Remember that the acceleration is given in terms of the change in velocity divided by the change in time:

Plugging in the numbers, your calculations look like this:

      Your acceleration was 3.8 mph/s. But that can’t be right! You may already see the problem here; take a look at the original definition of acceleration:

      Your final speed was 0 mph, and your original speed was 75 mph, so plugging in the numbers here gives you this acceleration:

      In other words, –3.8 mph/s, not +3.8 mph/s — a big difference in terms of solving physics problems (and in terms of law enforcement). If you accelerated at +3.8 mph/s rather than –3.8 mph/s, you’d end up going 150 mph at the end of 20 seconds, not 0 mph. And that probably wouldn’t make the cop very happy.

      Now you have your acceleration. You can turn off your calculator and smile, saying, “Maybe I was going a little fast, officer, but I’m very law-abiding. Why, when I heard your siren, I accelerated at –3.8 mph/s just in order to pull over promptly.” The policeman pulls out his calculator and does some quick calculations. “Not bad,” he says, impressed. And you know you’re off the hook.

      Accounting for direction

      

The sign of the acceleration depends on direction. If you slow down to a complete stop in a car, for example, and your original velocity was positive and your final velocity was 0, then your acceleration is negative because a positive velocity came down to 0. However, if you slow down to a complete stop in a car and your original velocity was negative and your final velocity was 0, then your acceleration would be positive because a negative velocity increased to 0.

      Looking at positive and negative acceleration

      When you hear that acceleration is going on in an everyday setting, you typically think that means the speed is increasing. However, in physics, that isn’t always the case. An acceleration can cause speed to increase, decrease, and even stay the same!

Acceleration tells you the rate at which the velocity is changing. Because the velocity is a vector, you have to consider the changes to its magnitude and direction. The acceleration can change the magnitude and/or the direction of the velocity. Speed is only the magnitude of the velocity.

      Here’s a simple example that shows how a simple constant acceleration can cause the speed to increase and decrease in the course of an object’s motion. Say you take a ball, throw it straight up in the air, and then catch it again. If you throw the ball upward with a speed of 9.8 m/s, the velocity has a magnitude of 9.8 m/s in the upward direction. Now the ball is under the influence of gravity, which, on the surface of the Earth, causes all free-falling objects to undergo a vertical acceleration of –9.8 m/s². This acceleration is negative because its direction is vertically downward.

      With this acceleration, what’s the velocity of the ball after 1.0 second? Well, you know that

      Rearrange this equation and plug in the numbers, and you find that the final velocity after 1.0 second is 0 meters/second:

      After 1.0 second, the ball has zero velocity because it’s reached the top of its trajectory, just at the point where it’s about to fall back down again. So the acceleration has actually slowed down the ball because it was going in the direction opposite the velocity.

      Now see what happens as the ball falls back down to Earth. The ball has zero velocity, but the acceleration due to gravity accelerates the ball downward at a rate of –9.8 m/s². As the ball falls, it gathers speed before you catch it. What’s its final velocity as you catch it, given that its initial velocity at the top of its trajectory is zero?

      The time for the ball to fall back down to you is just the same as the time it took to reach the top of its trajectory, which is 1.0 second, so you can find the final velocity for this part of the ball’s motion with this calculation:

So the final velocity is 9.8 meters/second directed straight downward. The magnitude of this velocity — that is, the speed of the ball — is 9.8 meters/second. The acceleration increases the speed of the ball as it falls because the acceleration is in the same direction as the velocity for this part of the ball’s trajectory.

      

When you work with physics problems, bear in mind that acceleration can speed up or slow down an object, depending on the direction of the acceleration and the velocity of the object. Don’t simply assume that just because something is accelerating its speed must be increasing. (By the way, if you want to see an example of how an acceleration can leave the speed of an object unchanged, take a look at the circular motion topic covered in Chapter 7.)

      Examining average and instantaneous acceleration