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Displacement, Velocity,
and Acceleration

Master the fundamental quantities of motion: how objects move, how fast they move, and how their speed changes over time.

AP Physics 1

What is Motion?

In physics, we describe motion using three closely related quantities. Each one answers a different question about a moving object:

Displacement
How far and in what direction did the object move from its starting point?
Velocity
How quickly is the object's position changing over time?
Acceleration
How quickly is the object's velocity changing over time?

Defining Displacement

Displacement is the change in position of an object. It is a vector that points from the initial position to the final position, regardless of the path taken.

Path taken A B Displacement \(\vec{\Delta x}\)

Distance vs Displacement

These two terms sound similar, but they describe fundamentally different things. Understanding the difference is critical for AP Physics.

Distance Displacement
Type Scalar Vector
Definition Total path length Change in position
Can be zero? Only if no motion Yes, if you return to start
Can be negative? No (always \(\ge 0\)) Yes (direction-dependent)
Symbol \(d\) \(\Delta x\)

Defining Velocity

Velocity is the rate of change of displacement with respect to time. Like displacement, it is a vector quantity.

Average Velocity
$$\vec{v}_{avg} = \frac{\Delta x}{\Delta t}$$
Instantaneous Velocity
$$\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$$

Speed vs Velocity

Just like distance vs displacement, speed and velocity are related but different:

Speed Velocity
Type Scalar Vector
Formula \(\frac{\text{distance}}{\Delta t}\) \(\frac{\Delta x}{\Delta t}\)
Can be negative? No Yes
Direction? No Yes

Defining Acceleration

Acceleration is the rate of change of velocity with respect to time. It tells you how quickly velocity is increasing or decreasing.

Average Acceleration
$$\vec{a}_{avg} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{\Delta t}$$
Instantaneous Acceleration
$$\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}$$

The Kinematic Equations

For constant acceleration, these four equations relate displacement, velocity, acceleration, and time:

1
\(v_f = v_i + at\)
2
\(\Delta x = v_i t + \frac{1}{2}at^2\)
3
\(v_f^2 = v_i^2 + 2a\Delta x\)
4
\(\Delta x = \frac{v_i + v_f}{2} \cdot t\)

Reading a Position-Time Graph

Time (s) Position (m) 2 4 6 20 40 60 Constant velocity At rest Accelerating

Reading a Velocity-Time Graph

Time (s) Velocity (m/s) 2 4 6 10 20 30 Accelerating Constant v Decelerating

How They Connect

Position \(x\) Velocity \(v = \frac{dx}{dt}\) Acceleration \(a = \frac{dv}{dt}\) derivative derivative integral integral

Solving Kinematics Problems

1
Draw a diagram. Sketch the situation with a coordinate axis and label the positive direction.
2
List knowns and unknowns. Identify \(x_i\), \(x_f\), \(v_i\), \(v_f\), \(a\), and \(t\). You need three knowns.
3
Choose the right equation. Pick the kinematic equation that contains your three knowns and the one unknown.
4
Solve algebraically first. Rearrange before plugging in numbers. Check signs carefully.
5
Check your answer. Does the sign make sense? Are the units correct? Is the magnitude reasonable?

Example 1: Finding Displacement

A car starts from rest and accelerates uniformly at \(a = 3\,\text{m/s}^2\) for \(t = 5\,\text{s}\). What is the car's displacement?

1
Knowns: \(v_i = 0\), \(a = 3\,\text{m/s}^2\), \(t = 5\,\text{s}\). Unknown: \(\Delta x\).
2
Equation: \(\Delta x = v_i t + \frac{1}{2}at^2\)
3
Substitute: \(\Delta x = (0)(5) + \frac{1}{2}(3)(5^2)\)
4
Solve: \(\Delta x = 0 + \frac{1}{2}(3)(25) = \) \(37.5\,\text{m}\)

Example 2: Finding Final Velocity

A ball is dropped from rest off a cliff. It falls \(\Delta x = 45\,\text{m}\). What is its final velocity? Use \(g = 10\,\text{m/s}^2\) downward.

1
Knowns: \(v_i = 0\), \(\Delta x = 45\,\text{m}\), \(a = 10\,\text{m/s}^2\). Unknown: \(v_f\).
2
Equation (no \(t\)): \(v_f^2 = v_i^2 + 2a\Delta x\)
3
Substitute: \(v_f^2 = 0 + 2(10)(45) = 900\)
4
Solve: \(v_f = \sqrt{900} = \) \(30\,\text{m/s}\) downward

Example 3: Braking Distance

A car is traveling at \(v_i = 20\,\text{m/s}\) when the driver brakes with an acceleration of \(a = -5\,\text{m/s}^2\). How far does the car travel before stopping?

1
Knowns: \(v_i = 20\,\text{m/s}\), \(v_f = 0\), \(a = -5\,\text{m/s}^2\). Unknown: \(\Delta x\).
2
Equation: \(v_f^2 = v_i^2 + 2a\Delta x\)
3
Rearrange: \(\Delta x = \frac{v_f^2 - v_i^2}{2a} = \frac{0 - 400}{2(-5)}\)
4
Solve: \(\Delta x = \frac{-400}{-10} = \) \(40\,\text{m}\)

Common Mistakes

✕ Ignoring Signs
Forgetting that velocity and acceleration can be negative. A negative acceleration does not always mean "slowing down."
✕ Confusing \(v\) and \(a\)
An object at zero velocity can still have nonzero acceleration (e.g., a ball at the top of its path).
✕ Using Equations When \(a\) Varies
Kinematic equations only work for constant acceleration. If \(a\) changes, you need calculus or graphs.

Question 1 of 5

A runner jogs \(200\,\text{m}\) east, then turns around and jogs \(50\,\text{m}\) west. What is the runner's displacement?

Question 2 of 5

An object moves with constant velocity \(v = 8\,\text{m/s}\) for \(t = 6\,\text{s}\). What is its acceleration?

Question 3 of 5

A ball is thrown straight up. At the very top of its path, which of the following is true?

Question 4 of 5

A car accelerates from \(v_i = 10\,\text{m/s}\) to \(v_f = 30\,\text{m/s}\) in \(t = 4\,\text{s}\). What is its average acceleration?

Question 5 of 5

On a velocity-time graph, the area under the curve between two times represents: