AP PHYSICS 1 AND 2 › UNIT 1, KINEMATICS › TOPIC 1.2
Displacement, Velocity, and Acceleration
Topic 1.1 gave you the quantities. This one puts a clock on them. Every definition below is a change divided by an interval, and every one of them throws away what happened in between.
1.2.A Describe a change in an object’s position.
1.2.B Describe the average velocity and acceleration of an object.
1. First, we throw away almost everything
Before any of this works we adopt the object model. The size, the shape and the internal structure of the thing are ignored, and what is left is a single point that still carries the properties we need, such as mass and charge.
That is a deliberate lie, and it is the most useful one in physics. A car becomes a point. A planet becomes a point. You stop asking which part of the car you are timing, and the question becomes answerable.
With that model in hand, displacement is the change in position:
Δx = x − x0
Two endpoints. Nothing else. The route is not in that expression anywhere.
2. Both averages are built from the ends alone
vavg = Δx / Δt aavg = Δv / Δt
Read those carefully. Average velocity is not the average of the velocities. It is the displacement divided by the time, and the object could have done anything at all in between. Two runners with the same average velocity over the same interval can have covered wildly different distances and reached wildly different top speeds.
On a graph of position against time, the average velocity over an interval is the slope of the straight line joining the two endpoints. That line is called a secant.
Shrink the interval
The curve is one object’s position against time. Drag either endpoint of the interval. The straight line is the secant, and its slope is the average velocity over what you have selected. Then shrink the interval and watch where the secant goes.
As the interval closes on a single instant, the secant becomes the tangent, and the average velocity becomes the instantaneous velocity at that moment. That is the whole idea behind 1.2.B.5, and in AP Physics C it is written as a derivative. Here it is enough to know that a short enough interval gives you a number very close to the instantaneous one.
3. “Accelerating” is a bigger word than you think
An object is accelerating if the magnitude of its velocity changes, or the direction of its velocity changes, or both. That word “or” does a great deal of work. A car going round a bend at a steady 20 m/s is accelerating the entire way, because the direction of its velocity is turning even though the speedometer never moves.
Along a single axis there is no direction left to change, so only the first clause survives, and the question becomes about signs:
Signs, and what they actually do
Choose a sign for the velocity and a sign for the acceleration. The dot moves under exactly those conditions. Watch the speed, not the position.
Notice what the simulation will not let you conclude: that a negative acceleration means slowing down. Two of the four cases have a negative acceleration, and in one of them the object speeds up the whole time.
4. A ball thrown straight up, at the top
At the highest point the velocity is momentarily zero. The acceleration is not. It is still g, still directed downward, still about 10 m/s2, and if it were ever zero the ball would hang there forever.
“The acceleration is zero because the ball has stopped” confuses a velocity with the rate at which velocity changes. The ball has stopped moving for an instant. It has not stopped changing its velocity, which is precisely why it does not stay stopped.
Check yourself
1. A runner is at x = 0 at t = 0, at +16 m at t = 4.0 s, and at +4 m at t = 8.0 s. Find the average velocity for each leg and for the whole trip.
Leg one: 16/4.0 = +4.0 m/s. Leg two: (4 − 16)/4.0 = −3.0 m/s. Whole trip: (4 − 0)/8.0 = +0.5 m/s. Note that the whole-trip answer is not the average of the two legs. It comes from the endpoints, and the endpoints only.
2. Two runners have the same average velocity over the same interval but covered different distances. Explain how, and say which one had the higher average speed.
Average velocity depends only on the two endpoints, so any two runners who start and finish in the same places at the same times share it. One of them may have wandered backward and forward on the way. That runner covered more distance, so with the same elapsed time the runner had the higher average speed.
3. Name a motion with constant speed and nonzero acceleration. Say which part of the definition of accelerating does the work.
Anything moving in a circle at a steady rate: a car on a roundabout, a satellite, a point on a spinning wheel. The magnitude of the velocity is constant, so the clause that does the work is the one about the direction of the velocity changing.
Topic 1.3 turns all of this into graphs and equations, where the slope of a position-time graph is a velocity and the area under a velocity-time graph is a displacement.