Velocity & Acceleration — Print · Grade 12 Science

Velocity & Acceleration

Learning goals

• I can distinguish average velocity, instantaneous velocity, and average speed, and calculate each.
• I can define acceleration as the rate of change of velocity and calculate it, including its sign.
• I can explain the meaning of g and apply it to objects in free fall.

Readings for this unit

• One ball, two component stories and a shared clock

Velocity is displacement over time, and the word 'average' hides an important subtlety. Average velocity is the total displacement divided by the total time, v_avg = Δx ÷ Δt, and because it is built on displacement it is a vector that can even come out to zero. A runner who circles a 400 m track and finishes back at the start has an average velocity of exactly zero for the lap, even though she was never once standing still — her displacement was zero, so her average velocity must be too. Average speed, by contrast, uses total distance instead of displacement, so for that same lap it is a healthy 8 m/s if the lap took 50 seconds.

Average velocity smooths a whole trip into one number, but often you want to know how fast something is moving *right now*. That is instantaneous velocity — the velocity at a single instant, which is what a car's speedometer shows (paired with the direction the car is pointing). Formally it is the slope of the position-time graph at that exact moment, the limit of Δx ÷ Δt as the time interval shrinks toward zero. On a road trip your average velocity might be 90 km/h, while your instantaneous velocity swings from 0 at a rest stop to 120 on the open highway.

Acceleration is the next step up: the rate at which velocity changes, a = (v − v₀) ÷ t, measured in meters per second per second, or m/s². Because velocity is a vector, a change in *either* its size or its direction counts as acceleration. A car speeding from 12 m/s to 30 m/s in 6 seconds accelerates at (30 − 12) ÷ 6 = 3 m/s². The sign matters and it is not just 'plus for go, minus for stop' — a negative acceleration means the velocity is becoming more negative, which slows an object moving in the positive direction but speeds up one already moving negative.

The most important acceleration in this unit is the one gravity gives for free. Near Earth's surface, every object in free fall — moving under gravity alone, with air resistance ignored — accelerates downward at the same rate, about 9.8 m/s², a value we call g. That phrase 'the same for every object' is the counterintuitive heart of it: a bowling ball and a marble, dropped together in a vacuum, gain speed at identical rates and land together. Galileo argued this four centuries ago, and astronauts have since dropped a hammer and a feather side by side on the airless Moon to prove it.

Free fall exposes a classic trap about the difference between velocity and acceleration. Throw a ball straight up and, at the very peak of its flight, its velocity is momentarily zero — for one instant it isn't moving up or down. But its acceleration there is *not* zero; it is still 9.8 m/s² downward, exactly as it is during the whole flight. Gravity never switches off, so even at the top the ball is changing its velocity, which is precisely why it doesn't hover but immediately begins to fall. Zero velocity and zero acceleration are completely different conditions.

Put together, these ideas let you describe motion with real precision. Average velocity captures the net result of a trip, instantaneous velocity captures a single moment, average speed tracks the total ground covered, and acceleration measures how the motion is changing — with g as the ready-made value for anything falling freely. Engineers designing everything from elevator ride-comfort to spacecraft re-entry work in exactly these terms, and a self-driving car's computer updates an object's instantaneous velocity and acceleration many times every second.

Model check: Negative acceleration need not mean slowing; compare its direction with velocity. a=(−8+2)/3=−2 m/s²; speed increases because velocity and acceleration point together.

Average velocity

v_avg = Δx ÷ Δt

Δx is displacement (a vector) and Δt is the time interval; a round trip has Δx = 0, so its average velocity is 0 even though distance was covered.

Average speed

average speed = total distance ÷ total time

a scalar built from path length, not displacement; it is never negative.

Acceleration

a = (v − v₀) ÷ t

v is the final velocity and v₀ the initial velocity (both in m/s), t is time in s, and a comes out in m/s². A negative a means the velocity is decreasing.

Free-fall acceleration

g ≈ 9.8 m/s²

the constant downward acceleration of any object in free fall near Earth's surface, independent of the object's mass.

Worked examples

Avoid these mistakes

• Negative acceleration need not mean slowing; compare its direction with velocity.

Words to know

average velocity

Displacement divided by the time interval: v_avg = Δx ÷ Δt. A vector, so it can be zero even after a long trip.

instantaneous velocity

The velocity at a single instant — the value on a speedometer paired with a direction; the slope of a position-time graph at that point.

average speed

Total distance divided by total time; a scalar that is never negative.

acceleration

The rate at which velocity changes: a = (v − v₀) ÷ t. A vector, measured in m/s².

free fall

Motion under the influence of gravity alone, with air resistance neglected.

g

The acceleration due to gravity near Earth's surface, about 9.8 m/s² directed downward, the same for every freely falling object regardless of mass.

Instantaneous velocity and the invention of calculus

Average velocity over a whole trip is easy, but 'how fast am I going at this exact instant?' is a surprisingly deep question — it asks for Δx ÷ Δt as the time interval shrinks toward zero, which is a division of nearly-nothing by nearly-nothing. Wrestling with exactly this problem led Isaac Newton and Gottfried Leibniz, working independently in the late 1600 s, to invent calculus. Instantaneous velocity is the derivative of position with respect to time, and instantaneous acceleration is the derivative of velocity. Your car's speedometer is, in effect, computing a derivative in real time — reading how fast position is changing at this very moment, not averaged over the trip.

Why g is the same for a bowling ball and a feather

It is genuinely strange that heavy and light objects fall at the same rate. A heavier object is pulled harder by gravity — but it also has more mass resisting that pull, and the two effects cancel exactly, leaving the same acceleration g for everything. This equivalence puzzled physicists for centuries until Einstein made it the cornerstone of general relativity. In everyday life air resistance hides the effect (a feather really does flutter down slowly), but remove the air and the trick vanishes: in a vacuum chamber, a feather and a bowling ball dropped together hit the bottom at the same instant, a demonstration you can watch physicists perform today.

The Leaning Tower legend and Galileo's real experiments

The famous story of Galileo dropping two balls from the Leaning Tower of Pisa is probably a myth, but the science behind it is real. Galileo couldn't time fast free fall with the clocks of his day, so he cleverly slowed gravity down by rolling balls down gentle ramps, where he could measure that the distance covered grew with the square of the time — the signature of constant acceleration. His conclusion overturned Aristotle's 2,000-year-old claim that heavier objects fall faster. Centuries later, in 1971, Apollo 15's David Scott closed the loop by dropping a hammer and a feather on the airless Moon: they landed together, on live television.

Negative acceleration is not the same as slowing down

A common misconception is that negative acceleration always means 'slowing down.' In truth, the sign of acceleration only tells you which way the velocity is changing, not whether speed is increasing or decreasing. What matters is how the acceleration's direction compares to the velocity's. A ball thrown upward has a negative (downward) acceleration the whole time — but on the way up that slows it down, while on the way down that speeds it up. An object speeds up whenever acceleration points the same way as velocity, and slows down when they point opposite ways, regardless of the raw sign. Careful sign bookkeeping, with a clearly chosen positive direction, is what keeps kinematics problems honest.

In the real world

A car's speedometer displays instantaneous speed; pairing it with the heading from a compass or GPS gives instantaneous velocity — exactly the quantity a self-driving car's computer updates dozens of times a second to decide when to brake or turn.

Exit check

v changes−2 to−8 m/s in 3 s. Find acceleration and speed change.