Topic 1.3 Representing Motion: translate between verbal, mathematical and graphical representations of motion, and interpret the slopes and areas of position-time, velocity-time and acceleration-time graphs.

What this topic is asking

The College Board (Topic 1.3) wants you to move fluently between the three ways physicists describe motion: in words, in equations, and in graphs. The graphical skill is the one the exam tests hardest, because question sets routinely hand you a position-time, velocity-time or acceleration-time graph and ask you to extract numbers and physical meaning from its slopes and areas.

The three representations

A single motion can be written three ways, and a strong answer can translate freely between them:

• Verbal: "the car starts from rest, speeds up steadily, then cruises at constant speed."
• Algebraic: the kinematic equations, such as $x = x_0 + v_0 t + \tfrac{1}{2}at^2$.
• Graphical: plots of position, velocity or acceleration against time.

Each representation highlights something different, and the exam expects you to pick the most useful one and convert when needed.

What slopes mean

So a position-time graph that is a straight line represents constant velocity, while a curved (parabolic) position-time graph represents accelerated motion. On a velocity-time graph, a horizontal line means constant velocity (zero acceleration), and a sloped straight line means constant acceleration.

What areas mean

Because AP Physics 1 uses constant-acceleration motion, the areas are simple geometric shapes: rectangles (constant velocity) and triangles (linearly changing velocity). Splitting a velocity-time graph into rectangles and triangles, finding each area, and adding them with the correct sign is the standard route to a displacement from a graph.

Reading curvature and sign

The shape of a position-time graph encodes both speed and direction. A line sloping up represents motion in the positive direction; a line sloping down represents motion in the negative direction; a horizontal line means the object is at rest. Curvature tells you about acceleration: a position-time graph that bends upward (concave up) has an increasing slope, so velocity is increasing, while a graph that bends downward has a decreasing slope. On a velocity-time graph, the sign of the velocity tells you the direction of motion and the sign of the slope tells you whether the object is speeding up or slowing down. When velocity and acceleration have the same sign the object speeds up; when they have opposite signs it slows down, which is exactly the rule from Topic 1.2 seen graphically. Being able to look at any one of the three graphs and sketch the other two, while keeping the signs consistent, is what separates a confident answer from a guess.

Try this

Q1. A position-time graph is a horizontal line. Describe the motion. [1 point]

• Cue. Zero slope means zero velocity, so the object is at rest.

Q2. An acceleration-time graph shows a constant $2.0$ m/s squared for $5.0$ s. Calculate the change in velocity. [2 points]

• Cue. Area $= (2.0)(5.0) = 10$ m/s increase in velocity.