AP® Physics C: Mechanics · Unit 1 · Topic 1.5
Two-Dimensional Motion with Vector Calculus
In two dimensions, position, velocity, and acceleration are vectors written with i and j components, and you get each one from the last by differentiating component by component.
Once motion leaves a straight line, one number can't describe position anymore, so you write r(t) = x(t) i + y(t) j. The i and j directions don't interact, so calculus treats them separately: differentiate the x(t) part to get vx(t), differentiate y(t) to get vy(t), and stack them back into a vector v(t) = vx(t) i + vy(t) j. Do it again and you get acceleration. Nothing new happens mathematically; you're just running the same derivative rules twice, once per axis.
The geometry is where it gets interesting. Velocity is always tangent to the path, showing the direction of travel at that instant. Acceleration doesn't have to point along the path at all. If acceleration has a component parallel to velocity, that part changes speed. If acceleration has a component perpendicular to velocity, that part changes direction, bending the path into a curve without speeding it up or slowing it down. An object speeding up along a curved path has acceleration tilted partly ahead of its velocity, not lined up with it, which is why the acceleration vector looks like it points off to one side.
This is the bridge from straight-line kinematics into everything curved that follows, including circular motion and projectiles. Speed and direction of travel come straight from the velocity vector's components: speed is its magnitude via the Pythagorean theorem, direction is its angle found with arctangent. The special case of acceleration perpendicular to velocity, changing direction only, is exactly the setup for uniform circular motion later in the course.
Key terms
- position vector
- A vector r(t) = x(t) i + y(t) j giving an object's location from the origin at time t.
- velocity vector
- The derivative of the position vector with respect to time, always tangent to the path of motion.
- acceleration vector
- The derivative of the velocity vector with respect to time, describing how velocity changes.
- speed
- The magnitude of the velocity vector, computed as √(vx2 + vy2).
- tangential and perpendicular components
- The parts of acceleration parallel to velocity, which change speed, and perpendicular to it, which change direction.
Watch it explained
- How it worksHow do you differentiate a position vector that is written with i and j components?
- Worked exampleFind the velocity and acceleration at t = 2 s for r(t) = 3t2 i + (t3 - 4t) j meters.
- Worked exampleFind the speed and direction of travel of a particle whose velocity is 6i - 8j m/s.
- Going furtherWhat does it mean about the motion when velocity and acceleration point in perpendicular directions?
Check yourself
End of topic.