you can use the equation Solving for
Plugging in the numbers and solving gives you the change in time:
Okay, so it takes 2.0 seconds for you to reach a speed of 62 m/s if your rate of acceleration is 31 m/s2. Now you can use this equation to find the total distance you need to travel to get up to this speed; it is the size of the displacement, which is given by
Plugging in the numbers gives you
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.
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 far, so good. Now you can plug this average velocity into the
And this becomes
You can also put in
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):
