Uniformly Accelerated Motion – Two Cars Meeting
PhysicsKinematicsAges 15–16
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Kinematics
Two cars, A and B, move along the same straight road with constant acceleration, each with its own initial position x₀, initial velocity v₀ and acceleration a (which may be negative). The x–t and v–t graphs are drawn as time runs, and the time and position at which the cars meet are marked. Change v₀, a and x₀, or drag on a graph to scrub through time, and compare with x = x₀ + v₀t + at²/2.
Lesson: Motion with constant acceleration, acceleration, velocity–time graphs
What it shows
Motion with constant acceleration is described by v = v₀ + at and x = x₀ + v₀t + ½at², where the signs of v₀ and a are measured along a chosen positive direction. On the x–t graph such motion is a parabola; on the v–t graph it is a straight line whose gradient is the acceleration and whose area gives the displacement. When v and a have opposite signs the car slows down, stops for an instant and then reverses. Two cars meet when their positions are equal, which appears as an intersection of their x–t curves and can be found by solving a quadratic equation.
How to use
Run the default case first: car A starts at the origin while car B, 200 m away, drives towards it and slows down. Predict when and where they meet, then compare with the marked crossing of the x–t curves. Next give one car an acceleration opposite in sign to its initial velocity so that it stops and reverses, and watch its v–t line cross the time axis. Drag on either graph to scrub back to any instant.
Parameters you can change
- Initial position of car A -100–100 m
- Initial velocity of car A -20–30 m/s
- Acceleration of car A -3–3 m/s²
- Initial position of car B 0–400 m
- Initial velocity of car B -30–20 m/s
- Acceleration of car B -3–3 m/s²
Questions to explore
- What do the gradient of the v–t graph and the area under it represent, and how can you check this here?
- Using the signs of v and a, how can you tell whether a car is speeding up or slowing down?
- Find a set of values for which the two cars meet twice. Why is this possible?
