Projectile Motion

The whole subject rests on one idea: horizontal and vertical motion are independent. Gravity acts only downward, so it changes the vertical velocity and leaves the horizontal velocity completely unchanged throughout the flight. The horizontal component v cos θ is constant from launch to landing; only the vertical component v sin θ is reduced by g, reaching zero at the apex and then reversing.

That independence explains the standard results. The time of flight depends purely on the vertical problem, T = 2v sin θ/g, and the range is simply the constant horizontal speed multiplied by that time, giving R = v² sin 2θ/g. Since sin 2θ is greatest at 2θ = 90 degrees, the maximum range is at 45 degrees.

It also follows that complementary angles give the same range: 30 and 60 degrees, or 20 and 70, land in the same place. The steeper shot trades horizontal speed for a longer hang time, and the two effects cancel exactly. The trajectory itself is a parabola, and at the apex the projectile is still moving, horizontally, at v cos θ. Note that mass never appears: in the absence of air resistance all projectiles follow the same path.

• Try this: Watch the horizontal velocity readout through a whole flight. Why does it never change, even at the apex?
• Try this: Set 30 degrees, note the range, then set 60. Why do two very different-looking trajectories land in the same place?
• Try this: Reduce gravity to lunar values. What happens to range and time of flight, and which one changes proportionally more?