AP Physics C: Mechanics · Unit 1 · Topic 6 of 6 · 6 min lesson

Vectors & 2D Motion

A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 6 practice questions.

Quiz this topic · 6 questions

The big idea

In two dimensions, treat each direction separately. Write position as r(t) = x(t) î + y(t) ĵ, then differentiate or integrate each component on its own. Combine them at the end for speed or direction.

1. Vector calculus, one component at a time

v = dr/dt = (dx/dt) î + (dy/dt) ĵvelocity

a = dv/dtacceleration, component by component

Speed = √(v_x² + v_y²)magnitude of v

Projectile: a_x = 0, a_y = −gno air resistance

The horizontal velocity never changes; gravity only changes the vertical part.

2. Projectiles

Split the launch velocity: v_0x = v₀ cos θ and v_0y = v₀ sin θ.

The time in the air comes from the vertical motion alone. The horizontal distance is v_x × time.

At the top, v_y = 0 but v_x doesn’t, so the speed there is v_x, the minimum speed. The acceleration (g, down) is perpendicular to the velocity there.

3. Example, then try it

Example

A particle’s position is r(t) = (4t) î + (3t² − t³) ĵ (meters). Find its velocity, speed and acceleration at t = 1 s.

v = 4 î + (6t − 3t²) ĵ, so v(1) = 4 î + 3 ĵ m/s and the speed is √(16 + 9) = 5 m/s.

a = (6 − 6t) ĵ, so a(1) = 0: the y-velocity is at its peak.

Try it · work it on paper first

A ball is launched at 20 m/s, 53° above level ground (sin 53° = 0.80, cos 53° = 0.60). Find the maximum height, the time in the air, the range and the speed at the top. (g = 10 m/s²)

Components: v_x = 20(0.60) = 12 m/s; v_0y = 20(0.80) = 16 m/s.

To the top: v_y = 16 − 10t = 0 at t = 1.6 s; height = (16)(1.6) − 5(1.6)² = 12.8 m.

Time in air: 2 × 1.6 = 3.2 s. Range: (12)(3.2) = 38.4 m.

Speed at the top: only v_x is left: 12 m/s.

Answer: 12.8 m high, 3.2 s, 38.4 m, 12 m/s at the top.

4. Common mistakes

“Speed is zero at the top.”Only v_y is zero; v_x remains.

Differentiating the magnitude instead of the components.Differentiate each component, then find the magnitude.

Mixing x and y in one equation.Keep the two directions separate until the end.

Check yourself

For a projectile with no air resistance, which quantity stays constant during the flight?

With no horizontal force, a_x = 0, so v_x never changes. Gravity changes v_y, and with it the speed and direction.

Ready to practice?

6 questions on Vectors & 2D Motion, each with a hint and an explanation for every wrong answer.

Quiz this topic All of Unit 1