Define and calculate displacement, average velocity, and acceleration, and distinguish each from the everyday words distance and speed (MA STE Introductory Physics, Motion and Forces).

What this topic is asking

This is the vocabulary of motion, and the Massachusetts Introductory Physics MCAS uses it constantly. You must define displacement, velocity, and acceleration, calculate each with the reference-sheet formulas, and keep them separate from the everyday words distance and speed. The crosscutting idea is using mathematics and computational thinking: almost every motion item asks you to substitute numbers into one of these relationships and report an answer with the right unit.

Displacement and distance

If you walk $10$ m east and then $4$ m west, the distance is $14$ m but the displacement is $6$ m east. On the MCAS, a back-and-forth or a round trip is the classic way to test this: a round trip has a large distance but a displacement of zero, because the start and end points are the same.

Velocity and speed

Average speed is the distance covered divided by the time taken, a scalar:

Average velocity is the displacement divided by the time taken, a vector with the same direction as the displacement. The reference sheet writes the average-velocity relationship as $v = \dfrac{d}{t}$, where $d$ is the displacement and $t$ is the time. Both are measured in meters per second (m/s).

The difference shows up whenever the path is not a straight line in one direction. A car that drives a $400$ m lap in $40$ s has an average speed of $10$ m/s but an average velocity of zero, because it returns to the start, so the displacement is zero.

Acceleration

The reference-sheet formula is

where $v_f$ is the final velocity, $v_i$ is the initial velocity, and $t$ is the time. The symbol $\Delta$ (delta) means change in. An object accelerates if it speeds up, slows down, or changes direction, because in each case the velocity is changing. A negative acceleration (sometimes called deceleration) means the velocity is decreasing or the object is accelerating in the negative direction.

The crucial point the MCAS tests: acceleration is about the change in velocity, not the velocity itself. A car moving at a steady $100$ km/h has a high speed but zero acceleration, because its velocity is not changing.

Reference-sheet note

The reference sheet gives the average-velocity relationship $v = \dfrac{d}{t}$ and the acceleration relationship $a = \dfrac{\Delta v}{\Delta t}$. It does not separately print "average speed," because in symbols it is identical to average velocity; you supply the understanding that one uses distance and the other uses displacement. Watch the units: velocity is m/s, acceleration is m/s squared.

Try this

Q1. A car covers a displacement of $150$ m in $10$ s. Calculate its average velocity. [2]

• Cue. $v = \dfrac{d}{t} = \dfrac{150}{10} = 15$ m/s.

Q2. State two ways an object can be accelerating even though its speed is constant. [1]

• Cue. It can change direction (for example moving in a circle); a change of direction is a change of velocity, so it is accelerating.