Physics I For Dummies. Steven Holzner

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      So it will take 62 meters of 31 m/s² acceleration to get you to takeoff speed — and the catapult is 100 meters long. No problem.

      Understanding uniform and nonuniform acceleration

      Acceleration can be uniform or nonuniform. Nonuniform acceleration requires a change in acceleration. For example, when you’re driving, you encounter stop signs or stop lights often, and when you slow to a stop and then speed up again, you take part in nonuniform acceleration.

      Other accelerations are very uniform (in other words, unchanging), such as the acceleration due to gravity near the surface of the Earth. This acceleration is 9.8 meters per second² downward, toward the center of the Earth, and it doesn’t change (if it did, plenty of people would be pretty startled).

Relating Acceleration, Time, and Displacement

      This chapter deals with four quantities of motion: acceleration, velocity, time, and displacement. You work the standard equation relating displacement and time to get velocity:

      And you see the standard equation relating velocity and time to get acceleration:

      But both of these equations only go one level deep, relating velocity to displacement and time and acceleration to velocity and time. What if you want to relate acceleration to displacement and time? This section shows you how you can cut velocity out of the equation.

      

When you’re slinging around algebra, you may find it easier to write single quantities like v (to stand for ) rather than . You can usually turn v into later if necessary.

      Not-so-distant relations: Deriving the formula

      You relate acceleration, displacement, and time by messing around with the equations until you get what you want. First, note that displacement equals average velocity multiplied by time:

      You have a starting point. But what’s the average velocity? If your acceleration is constant, your velocity increases in a straight line from 0 to its final value, as Figure 3-4 shows.

      FIGURE 3-4: Increasing velocity under constant acceleration.

      The average velocity is half the final velocity, and you know this because there’s constant acceleration. Your final velocity is , so your average velocity is half this:

      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):

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