Motion in a plane

Position and velocity in two dimensions

Narrated video. Transcript:

A paper plane glides across the classroom. To describe where it is, we now need two numbers at every instant: x and y.

Its position vector from the origin is r = x î + y ĵ. As the plane moves, watch the tip of r trace out its path.

In a short time Δt the plane moves from P to P′. Its displacement is Δr = r′ − r = Δx î + Δy ĵ.

The average velocity is displacement divided by time: v̄ = Δr / Δt. It points the same way as Δr.

Now make Δt smaller and smaller. Watch P′ slide towards P. The displacement arrow turns until it lies along the tangent to the path.

In the limit we get the instantaneous velocity, v = dr/dt. It always points along the tangent to the path, in the direction of motion.

In components, v = vx î + vy ĵ, where vx = dx/dt and vy = dy/dt. Its magnitude is √(vx² + vy²), and its direction is given by tan θ = vy / vx.

Acceleration in two dimensions

Narrated video. Transcript:

Velocity can change in size, in direction, or both. Any of these changes means there is an acceleration.

Average acceleration is the change in velocity divided by the time taken: ā = Δv / Δt. Place v and v′ tail to tail. The arrow from the tip of v to the tip of v′ is Δv.

Shrink the time interval and you get the instantaneous acceleration, a = dv/dt. In components, ax = dvx/dt and ay = dvy/dt.

Here is the big difference from one dimension. On a straight line, a and v always lie along the same line. In a plane, the angle between them can be anything from 0° to 180°.

A train speeding up on a straight track: a is along v. Braking: a is opposite to v. A ball whirled in a circle at steady speed: a is at right angles to v.

And a ball thrown across the field: a points straight down the whole time, while v keeps turning along the curved path.

Constant acceleration

Narrated video. Transcript:

Now let the acceleration a stay constant: the same size and the same direction at every moment.

Then the velocity changes steadily: v = v₀ + a t. Watch v₀ and the growing arrow a t join head to tail to make v.

The position follows r = r₀ + v₀t + ½at². Think of it as three arrows head to tail: start at r₀, add v₀t, then add ½at².

Each vector equation is really two ordinary equations, one for x and one for y: x = x₀ + v₀x t + ½ax t², and y = y₀ + v₀y t + ½ay t².

So motion in a plane is just two straight-line motions happening at the same time, each one independent of the other.

If v₀ and a lie along the same line, the path is straight. If they are at an angle, the path curves. A thrown cricket ball, with a pointing straight down, is the classic example.

Example: a particle starts at the origin moving at 4 m/s along x, with an acceleration of 2 m/s² along y. Then x = 4t and y = t².

At t = 3 s it is at x = 12 m, y = 9 m, and its velocity is 4î + 6ĵ m/s. The path is a parabola.

Try it: constant acceleration in a plane

Interactive simulation. Things to try:

• Press Play, or drag the time slider to move through the motion.
• Blue is the velocity, amber the acceleration. Watch v turn towards a as time goes on.
• Turn on the arrows for r₀ + v₀t + ½at² and see them add up to r.
• Press “Gravity only” to turn the motion into a projectile.

Quick check

A particle starts from the origin at t = 0 with velocity 5.0 ĵ m/s and moves in the x-y plane with a constant acceleration of (6.0 î + 2.0 ĵ) m/s². What is its y-coordinate when its x-coordinate is 27 m?

1. 15 m
2. 24 m
3. 27 m
4. 33 m

Think about it

A body moves in a plane with a constant acceleration. Which statement is always true?

1. Its path must be a straight line.
2. Its velocity always points the same way as its acceleration.
3. Its x and y motions can be treated separately, each with constant acceleration.
4. Its speed must keep increasing.

Summary

• Position: r = x î + y ĵ
• Velocity: v = dr/dt = vx î + vy ĵ
• Acceleration: a = dv/dt = ax î + ay ĵ
• Constant acceleration: v = v₀ + at, r = r₀ + v₀t + ½at²
• Along x (same for y): x = x₀ + v₀x t + ½ax t²

• The velocity always points along the tangent to the path.
• In a plane, the acceleration can be at any angle to the velocity, not just along it.
• With constant acceleration, the x and y motions are independent: solve each like a straight-line problem.
• The path is a straight line only if v₀ is along a (or zero). Otherwise it is a parabola.