Velocity and acceleration in a plane

In a plane, a body's place is given by its position vector r = x î + y ĵ, drawn from the origin. If it moves from r to r′, its displacement is Δr = r′ – r, the arrow from the old place to the new one.

The words from Chapter 2 carry over, now as vectors:

• Average velocity = Δr/Δt. Velocity v = dr/dt = vₓ î + v_y ĵ, which always points along the tangent to the path.
• Acceleration a = dv/dt. In a plane, v and a can make any angle from 0° to 180°: along v the body speeds up, against v it slows down, and at an angle it turns.

The trick of this chapter: the x-motion and the y-motion are independent. Treat each as a straight-line motion of Chapter 2. The only thing they share is time.

When a is constant, the Chapter 2 equations work along each axis:

• vₓ = v₀ₓ + aₓt and x = x₀ + v₀ₓt + ½aₓt²
• v_y = v₀ᵧ + a_yt and y = y₀ + v₀ᵧt + ½a_yt²

Example: a puck starts at the origin with velocity 5 î m/s, and a steady push gives it a = 2 ĵ m/s². After 3 s: x = 5 × 3 = 15 m (no x-acceleration), y = ½ × 2 × 3² = 9 m, and v = 5 î + 6 ĵ, of size √61 ≈ 7.8 m/s. Along x it never sped up at all.

So the method is always the same: solve the axis where you know enough to find t, then use that same t on the other axis.

As before, the average speed (path length ÷ time) is never less than the size of the average velocity; they are equal only on a straight path without turning back.