Physics I For Dummies. Steven Holzner

Physics I For Dummies

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Busting out the stopwatch: Average speed

Average speed is the total distance you travel divided by the total time it takes. Average speed is sometimes written as math

; a bar over a variable means average in physics terms.

Say, for example, that you want to drive from New York City to Los Angeles to visit your uncle’s family, a distance of about 2,781 miles. If the trip takes you 4.000 days, what was your average speed? You divide the total distance by the change in time, so your average speed for the trip would be

This solution divides miles by days, so you come up with 695.3 miles per day. Not exactly a standard unit of measurement — what’s that in miles per hour? To find it, you want to cancel days out of the equation and put in hours (see Chapter 2). Because a day is 24 hours, you can multiply this way (note that days cancels out, leaving miles over hours, or miles per hour):

That’s a better answer.

You can relate total distance traveled, s, with average speed, math

, and time, t, like this:

Contrasting average and instantaneous speed

Average speed differs from instantaneous speed, unless you’re traveling in uniform motion (in which case your speed never varies). In fact, because average speed is the total distance divided by the total time, it may be very different from your instantaneous speed.

If you travel 2,781 miles in 4.000 days (a total of 96 hours), you go at an average speed of 28.97 miles per hour. That answer seems pretty slow, because when you’re driving, you’re used to going 65 miles per hour. You’ve calculated an average speed over the whole trip, obtained by dividing the total distance by the total trip time, which includes non-driving time. You may have stopped at a hotel several nights, and while you slept, your instantaneous speed was 0 miles per hour; yet even at that moment, your overall average speed was still 28.97 miles per hour!

Distinguishing average speed and average velocity

There is a difference between average speed and average velocity. Say, for example, that while you were driving in Ohio on your cross-country trip, you wanted to make a detour to visit your sister in Michigan after you stopped by a friend’s house in Indiana. Your travel path may have looked like the straight lines in Figure 3-3 — first 80 miles to Indiana and then 30 miles to Michigan.

Schematic illustration of a trip from Ohio to Michigan.

FIGURE 3-3: A trip from Ohio to Michigan.

If you drove at an average speed or a uniform speed of 55 miles per hour and you had to cover math , this trip took you 2.0 hours. But if you calculate the magnitude of the average velocity (by taking the distance between the starting point and the ending point, about 85 miles as the crow flies), you get

The direction of the average velocity is just the direction between the start and end points. But if you’re interested in your average speed along either of the two legs of the trip, you have to measure the time it takes for a leg and divide the length of that leg by that time to get the average speed.

To calculate the average speed over the whole trip, you look at the whole distance traveled, which is math , not just 85 miles. And 110 miles divided by 2.0 hours is 55 miles per hour; this is your average speed.

As another illustration of the difference between average speed and average velocity, consider the motion of the Earth around the sun. The Earth travels in its nearly circular orbit around the sun at an enormous average speed of something like 18 miles per second! However, if you consider one full revolution of the Earth, the Earth returns to its original position, relative to the sun, after one year. After one year, there’s no displacement relative to the sun, so the Earth’s average velocity over a year is zero, even though its average speed is enormous!

When considering motion, it’s not only speed that counts but also direction. That’s why velocity is important: It lets you record an object’s speed and its direction. Pairing speed with direction enables you to handle cases like cross-country travel, where the direction can change.

Speeding Up (Or Down): Acceleration

Acceleration is a measure of how quickly your velocity changes. When you pass a parking lot’s exit and hear squealing tires, you know what’s coming next — someone is accelerating to cut you off. After he passes, he slows down right in front of you, forcing you to hit your brakes to slow down yourself. Good thing you know all about physics.

You may think that, with all this speeding up and slowing down, you’d use terms like acceleration and deceleration. Well, physics has no use for the term deceleration, because deceleration is just a particular kind of acceleration — one in which speed reduces.

Like speed, acceleration takes many forms that affect your calculations in various physics situations. In different physics problems, you have to take into account the direction of the acceleration (whether the acceleration is positive or negative in a particular direction), whether it’s average or instantaneous, and whether it’s uniform or nonuniform. This section tells you more about acceleration and explores its various forms.

Defining acceleration

In physics terms, acceleration, a, is the amount by which your velocity changes in a given amount of time, or

Given the initial and final velocities, v_i and v_f, and the initial and final times over which your speed changes, t_i and t_f, you can also write the equation like this:

Acceleration, like velocity, is actually a vector and is often written as a, in vector style (see Chapter 4). In other words, acceleration, like velocity but unlike speed, has a direction associated with it.

Determining the units of acceleration

You

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