Projectile Motion

You’ll be able to

Treat horizontal and vertical motion as independent
Analyze a horizontally launched projectile using time as the shared variable
Explain why horizontal velocity stays constant while vertical velocity changes

Two motions at once, handled separately

A projectile is any object moving under gravity alone once launched. Its power move is that the horizontal and vertical motions are independent. Horizontally there is no acceleration, so velocity is constant (Δx = v_x·t). Vertically there is only gravity, so it is a constant-acceleration problem with a = g downward. You solve the two axes as separate 1-D problems.

Time is the bridge between the axes

The single quantity the two axes share is time — the clock runs the same for both. So the usual strategy is to use the vertical equations to find how long the object is in the air, then feed that time into the horizontal equation to get range. For an object launched horizontally, the initial vertical velocity is zero.

Projectile equations (horizontal launch, taking down as +)

horizontal: Δx = v_x·t · vertical: Δy = ½g·t² · v_y = g·t

No acceleration horizontally (v_x constant); a = g ≈ 10 m/s² vertically.

Why a dropped ball and a thrown ball land together

Because the axes are independent, the horizontal velocity has no effect on how fast something falls. A ball dropped from a table and a ball rolled off the same table with high horizontal speed hit the floor at the same instant — gravity gives both the same downward acceleration, and their identical vertical drop takes identical time.

Worked example

A ball is thrown horizontally at 15 m/s from a 20 m high cliff. Using g = 10 m/s², how long is it airborne and how far from the base does it land?

1.Vertical first — find the fall time from Δy = ½g·t²: 20 = ½(10)t² = 5t².
2.Solve: t² = 20 ÷ 5 = 4, so t = 2 s.
3.Horizontal uses that shared time: Δx = v_x·t = (15 m/s)(2 s).
4.Evaluate: Δx = 30 m.

Answer: t = 2 s and the ball lands 30 m from the base

Tip

Always solve the vertical axis first for a horizontally launched projectile: it hands you the flight time with no dependence on horizontal speed. Then the horizontal distance is a one-step multiplication.

Checkpoint

From the same height, one ball is dropped and another is thrown horizontally at 12 m/s at the same instant. Which lands first? (Ignore air resistance.)

Checkpoint

A rock is thrown horizontally at 8 m/s from a cliff and stays in the air for 3 s. How far horizontally does it travel? (g = 10 m/s²)

Checkpoint

A ball is thrown horizontally at 10 m/s. Two seconds later, what is its horizontal velocity? (g = 10 m/s², ignore air resistance)

On the exam

The classic AP trap: applying gravity to the horizontal axis. Horizontal velocity is constant for every projectile. Gravity changes only the vertical velocity. Keep the two axes in separate columns on your paper.

Answer the 3 checkpoints as you read.