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Representing
Motion

Master how physicists describe movement using position, velocity, and acceleration through graphs, equations, and diagrams.

AP Physics 1

What Is Motion?

Motion is a change in position of an object with respect to time and a reference point. In physics, we don't just say something "moved": we specify how far, in what direction, and how quickly.

Reference Frame
The coordinate system from which motion is measured: an origin and axes
Position
Where an object is relative to the origin, described by \( x \) (in meters)
Time
The independent variable \( t \) (in seconds) that tracks when events occur

Scalars vs. Vectors

Every quantity describing motion is either a scalar (magnitude only) or a vector (magnitude and direction). Confusing the two is one of the most common early mistakes in physics.

Scalar Vector
Definition Magnitude only Magnitude + direction
Distance / Displacement Distance \( d \) Displacement \( \vec{\Delta x} \)
Speed / Velocity Speed \( |v| \) Velocity \( \vec{v} \)
Example "5 m/s" "5 m/s east"

Understanding Displacement

-2 -1 0 1 2 3 4
\( x_i = -1 \text{ m} \)
\( x_f = 3 \text{ m} \)
\( \Delta x = +4 \text{ m} \)
 Position Number Line (meters)

Average Velocity

Average velocity tells us the rate of change of position over a time interval. Unlike speed, velocity can be negative (indicating direction).

Average Acceleration

Acceleration measures how velocity changes over time. An object can accelerate by speeding up, slowing down, or changing direction.

Speeding Up
\( v \) and \( a \) have the same sign
Slowing Down
\( v \) and \( a \) have opposite signs

The Kinematic Equations

For constant acceleration in one dimension, these four equations connect position, velocity, acceleration, and time.

1
\( v = v_0 + at \) : velocity from acceleration and time
2
\( x = x_0 + v_0 t + \frac{1}{2}at^2 \) : position from time
3
\( v^2 = v_0^2 + 2a(x - x_0) \) : velocity without time
4
\( x = x_0 + \frac{1}{2}(v_0 + v)t \) : position from average velocity

Reading Position-Time Graphs

t (s) x (m) 2 4 6 2 4 6 0 Constant v At rest Accelerating

Reading Velocity-Time Graphs

t (s) v (m/s) 2 4 6 +3 0 -3 Constant a Constant v (a = 0)
Slope
The slope of a \( v \)-\( t \) graph equals acceleration
Area
The area under the curve equals displacement

Distance vs. Displacement

A runner goes 3 m east, then 1 m west 0 1 2 3 Start End 3 m east 1 m west Distance = 4 m Displacement = +2 m

Solving Kinematics Problems

1
Draw a diagram. Sketch the situation and choose a positive direction.
2
List knowns and unknowns. Write down \( x_0, x, v_0, v, a, t \).
3
Choose the right equation. Pick the kinematic equation that contains your unknown and your knowns.
4
Solve algebraically before plugging in numbers. Check units at the end.
5
Verify the sign. Does the direction make physical sense?

Example 1: Finding Velocity

A car starts from rest and accelerates at \( a = 3 \text{ m/s}^2 \) for \( t = 5 \text{ s} \). What is its final velocity?

1
Knowns: \( v_0 = 0 \), \( a = 3 \text{ m/s}^2 \), \( t = 5 \text{ s} \)
2
Unknown: \( v = \,? \)
3
Equation: \( v = v_0 + at \)
4
Solve: \( v = 0 + (3)(5) = \) \( 15 \text{ m/s} \)

Example 2: Finding Displacement

A bike moving at \( v_0 = 8 \text{ m/s} \) decelerates at \( a = -2 \text{ m/s}^2 \). How far does it travel before stopping?

1
Knowns: \( v_0 = 8 \text{ m/s} \), \( v = 0 \), \( a = -2 \text{ m/s}^2 \)
2
Unknown: \( \Delta x = \,? \) (no time given)
3
Equation: \( v^2 = v_0^2 + 2a\Delta x \)
4
Solve: \( 0^2 = 8^2 + 2(-2)\Delta x \implies 0 = 64 - 4\Delta x \implies \Delta x = \frac{64}{4} = \) \( 16 \text{ m} \)

Example 3: Graph Analysis

t (s) v (m/s) 2 4 6 0 2 4 6
\( a > 0 \)
\( a = 0 \)
\( a < 0 \)

Common Mistakes

✕ Ignoring Signs
Treating velocity and acceleration as always positive. A negative \( a \) doesn't always mean "slowing down": it means acceleration points in the negative direction.
✕ Mixing Distance and Displacement
Using total path length when the problem asks for displacement, or vice versa. Always check what the question is asking.
✕ Using Equations When \( a \) Isn't Constant
The kinematic equations only work for constant acceleration. If \( a \) changes, use graph-based methods (area under curves).

Motion Representations Summary

Graph Type Slope Gives Area Gives
\( x \)-\( t \) Velocity \( v \) Not typically used
\( v \)-\( t \) Acceleration \( a \) Displacement \( \Delta x \)
\( a \)-\( t \) Jerk (not in AP 1) Change in velocity \( \Delta v \)

Question 1 of 5

A car travels 60 m east then 20 m west. What is its displacement?

Question 2 of 5

On a position-time graph, what does the slope of the line represent?

Question 3 of 5

An object starts at rest and accelerates at \( 4 \text{ m/s}^2 \) for \( 3 \text{ s} \). What is its final velocity?

Question 4 of 5

An object moves with positive velocity and negative acceleration. The object is:

Question 5 of 5

The area under a velocity-time graph represents: