Motion in One Dimension
Position, velocity and acceleration, and why a graph's slope tells you something its height never will.
Introduction
A cart on a straight track: where is it, how fast is it going, and how is that changing?
A cart moves to the right. Its acceleration is a constant −2 m/s². What does it do over the next ten seconds?
Speeding up and slowing down
A cart speeds up when its velocity and acceleration point the same way, and slows down when they point opposite ways. The sign of the acceleration alone says nothing about speeding up.
Velocity is a slope
The slope of the position graph at an instant is the velocity at that instant. And, as your sketch showed, the slope of the velocity graph is the acceleration.
Reading the graphs
Each dashed target was drawn by this simulation with settings you cannot see.
Two carts move on side-by-side tracks. Their position–time graphs cross at t = 3 s. What must be true at t = 3 s?
Displacement and distance
The cart travelled 40.5 m and ended where it started. Those are two different quantities. Distance is how far it travelled. Displacement is where it ends up relative to where it started: here, zero.
The cart started and ended at −8 m, 9 seconds apart. What is its average velocity over those 9 seconds?
Displacement from a velocity graph
A cart's velocity falls steadily from 6 m/s to −2 m/s over 4 s. What is its displacement over those 4 s, and how far does it travel?
The constant-acceleration equations
When the acceleration is constant, and only then, two equations describe the motion. The first says the velocity changes by every second. The second is the parabola you just matched.
With air resistance, a spring or an orbit the acceleration changes, and these equations give wrong answers.
Stopping distance
A cart moving at 12 m/s brakes at a steady 3 m/s². How far does it go before it stops?
Practice
Each problem has its own numbers, and its answer comes from the simulation's model.
A cart starts at m with velocity m/s and a constant acceleration of m/s². Where is it at s?
A cart moving right at m/s has a constant acceleration of m/s². What is its velocity 5 s later?
A cart starts at m moving right at m/s, with an acceleration of m/s². Where does it turn around?
A cart starts moving right at m/s with an acceleration of m/s². What is its average velocity over the first 6 s?
A cart starts moving right at m/s with an acceleration of m/s². How far does it travel in the first 6 s? (Distance, not displacement.)
A cart starts at m moving right at m/s and speeds up at m/s². How fast is it moving as it passes ?
What comes next
You made a cart slow down, stop and turn round without changing its acceleration, and read its motion from the slopes of its graphs. Next the cart leaves the track: a projectile moves in two dimensions at once, and its horizontal and vertical motions go on independently of each other.
Ideas this lesson took on
Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.
- “Negative acceleration means slowing down”
- “A position–time graph is a picture of the path”
- “Distance and displacement are the same thing”
- “Where two position graphs cross, the velocities are equal”
Finished with this lesson?
Progress is stored on this device only. There is no account and nothing is sent anywhere.
Research behind this lesson
- Beichner 1994: Students routinely read a graph's height when its slope is what matters, and treat a position–time graph as a picture of the path the object took.
- Trowbridge & McDermott 1980: In demonstration interviews, students judged which of two objects was faster by which was ahead, reading velocity off position, with the confusion surfacing exactly when one object passes the other.
- Trowbridge & McDermott 1981: The companion interviews on acceleration: students routinely read acceleration off the velocity, momentarily at rest means no acceleration, moving faster means accelerating, rather than off its rate of change.
- McDermott et al. 1987: Connecting a graph to the physical situation it describes is a distinct skill, and it is not acquired as a by-product of learning either one alone.
- Sokoloff & Thornton 1997: A demonstration only teaches when students commit a written prediction, discuss it, watch the real event, and compare: a cycle that produced far larger gains on the Force and Motion Conceptual Evaluation than traditional lecture instruction.
- Van Heuvelen 1991: Experts move fluently between words, pictures, diagrams, graphs and equations; novices tend to jump straight to equations. Explicit practice moving between representations is what closes the gap.