Conceptual Physics

2.12 - Interactive checkpoint: subway train

A subway train accelerates along a

straight track at a constant 1.90 m/s^2.

How long does it take the train to

increase its speed from 4.47 m/s to

13.4 m/s?

Answer:

ǻt = s

2.13 - Interactive problem: what’s wrong with the rabbits?

You just bought five rabbits. They were supposed to be constant acceleration rabbits, but you worry that some are the less expensive, non-constant acceleration rabbits. In fact, you think two might be the cheaper critters. You take them home. When you press GO, they will run or jump for five seconds (well, one just sits still) and then the simulation stops. You can press GO as many times as you like and use the PAUSE button as well. Your mission: Determine if you were ripped off, and drag the “½ off” sale tags to the cheaper rabbits. The simulation will let you know if you are correct. You may decide to keep the cuddly creatures, but you want to be fairly charged. Each rabbit has a velocity gauge that you can use to monitor its motion in the simulation. The simplest way to solve this problem is to consider the rabbits one at a time: look at a rabbit’s velocity gauge and determine if the velocity is changing at a constant rate. No detailed mathematical calculations are required to solve this problem. If you find this simulation challenging, focus on the relationship between acceleration and velocity. With a constant rate of acceleration, the velocity must change at a constant rate: no jumps or sudden changes. Hint: No change in velocity is zero acceleration, a constant rate.

2.14 - Derivation: creating new equations

Other sections in this chapter introduced some of the fundamental equations of motion. These equations defined fundamental concepts; for example, average velocity equals the change in position divided by elapsed time. Several other helpful equations can be derived from these basic equations. These equations enable you to predict an object’s motion without knowing all the details. In this section, we derive the formula shown in Equation 1, which is used to calculate an object’s final velocity when its initial velocity, acceleration and displacement are known, butnot the elapsed time. If the elapsed time were known, then the final velocity could be calculated using the definition of velocity, but it is not. This equation is valid when the acceleration is constant, an assumption that is used in many problems you will be posed. Variables

We use t instead of ǻt to indicate the elapsed time. This is simpler notation, and we will use it often.

Deriving a motion equation

vf^2 = vi^2 + 2aǻx

vi = initial velocity

vf = final velocity

a = constant acceleration

ǻx = displacement

acceleration (constant) a

initial velocity vi

final velocity vf

elapsed time t

displacement ǻx