Before we can describe how something moves, we need to decide how to represent it. Physics uses something called the object model: we pretend the entire object is just a single point in space, with no size or shape.
Think about tracking a car driving across town. Does it matter that the car has four wheels, a back seat, and a trunk? Not for describing its motion from A to B. We just need to know where the car is at each moment. So we shrink it to a point and track that point.
The object still has properties that matter — its mass and charge, for example. We keep those. We just stop worrying about size and shape.
When something moves, its position changes. We use the symbol x for position, and we need a reference point — a zero — to measure from. That zero is called the origin.
Displacement is simply the change in position. It answers the question: how far did the object move, and in which direction?
Here, x is the final position and x₀ is the starting position. The Greek letter delta (Δ) always means "change in" — final minus initial.
Displacement is a vector quantity — it has both size and direction. In one dimension, direction is encoded in the sign: positive means one direction (usually right or up), negative means the other. This is not optional — getting the sign wrong gets the physics wrong.
A dog starts at position x₀ = +3 m, runs to x = −2 m, then back to x = +1 m. What is the dog's displacement? What distance did it travel?
Now that we can describe position changes, we want to describe how fast those changes happen. Average velocity answers: over some period of time, how much displacement happened per second?
The bar over the v (v̄) means "average." Δt is the time interval — how long the motion took. The result tells you how many meters of displacement occurred per second, on average, during that interval.
On a position vs. time graph, average velocity is the slope of the straight line connecting two points. That's it. Rise over run. Δx over Δt. Steeper slope = faster average velocity.
Hover over any point to read its value. The dashed green line shows average velocity — it's the slope from start to end.
A student walks from x₀ = 2 m to x = 14 m in 4 seconds, then turns around and walks back to x = 12 m in 1 more second. What is the student's average velocity for the entire 5-second trip?
Velocity can change over time. When it does, we say the object is accelerating. Average acceleration measures how quickly velocity changes — it's the velocity version of what velocity is to position.
Here, Δv is the change in velocity (v − v₀) and Δt is the time interval. The units are meters per second per second, written m/s². That means: for every second that passes, velocity changes by this many m/s.
On a velocity vs. time graph, average acceleration is the slope of the line connecting two velocity values. Same idea as before — just applied to velocity instead of position.
Hover any point to read velocity. The dashed amber line is average acceleration — slope from first to last velocity.
A car is traveling at v₀ = +20 m/s. It brakes and slows to v = +8 m/s over 4 seconds. What is the car's average acceleration?
This is where most students — and many teachers — get tripped up. Read this section slowly.
Speed is how fast something is moving — just a number, always positive. Velocity is speed with direction — it can be positive or negative. Acceleration responds to changes in velocity, not just speed.
A car goes from 0 m/s to 30 m/s heading east. Velocity changed in magnitude. This is acceleration.
That same car hits the brakes. It's still moving east, but slowing. Velocity is still positive but decreasing. This is also acceleration — a negative one.
A ball rolls north at 4 m/s, bounces off a wall, and rolls south at 4 m/s. Speed didn't change at all. But velocity went from +4 to −4. That's a massive change in velocity — huge acceleration during the bounce.
A ball rolls left at −6 m/s and hits a rough patch, slowing to −2 m/s over 2 seconds. Is the acceleration positive or negative?
Average velocity tells us what happened over an interval. But what if we want to know the velocity at one exact moment? That's instantaneous velocity.
Here's the key idea: if you make the time interval smaller and smaller, the average velocity gets closer and closer to the instantaneous velocity at that moment. In the limit — as Δt approaches zero — average velocity becomes instantaneous velocity.
Using the position function x = t², watch what happens to average velocity as the time interval shrinks around t = 2 s:
The true instantaneous velocity at t = 2 s is exactly 4.0 m/s. As Δt shrinks, the average gets closer and closer.
The same logic applies to acceleration: average acceleration calculated over a very small time interval gives a value very close to the instantaneous acceleration at that moment.
A runner's position is described by x = t² (in meters, with t in seconds). Estimate the instantaneous velocity at t = 3 s by using a very small time interval.