Projectile motion

A projectile is anything thrown into the air and then left to gravity: a ball, a javelin, water from a hose. It follows a curved path because its starting velocity and its acceleration (g, downward) are at an angle. (If they were along the same line, at 0° or 180°, it would just move in a straight line.)

Split the motion in two, as in the last lesson, ignoring air:

• Horizontal: no force, so no acceleration. The horizontal velocity u cos θ stays the same from launch to landing.
• Vertical: a = –g the whole time, exactly like a ball thrown straight up at u sin θ.

At the top, the vertical velocity is zero, but the body is still moving sideways at u cos θ. So the velocity at the top is not zero; it is horizontal.

From these come the three results for a throw from flat ground:

Example: a ball kicked at 20 m/s, 30° up (g = 9.8 m/s²). u sin θ = 10 m/s and u cos θ ≈ 17.3 m/s. T = 20/9.8 ≈ 2.04 s; H = 100/19.6 ≈ 5.1 m; R = 17.3 × 2.04 ≈ 35.3 m.

Two facts about the range:

• sin 2θ is largest at 2θ = 90°, so the greatest range is at 45°: R_max = u²/g. At 14 m/s, R_max = 196/9.8 = 20 m.
• θ and 90° – θ give the same range (15° and 75°, say): the steep throw goes higher and stays up longer, the flat one goes faster along the ground.

Take time out of x = (u cos θ)t and y = (u sin θ)t – ½gt², and you get the path: y = x tan θ – gx²/(2u² cos² θ). Only x is squared, so the path is a parabola.