Projectile Motion (Trajectory Parabola)

The fastest way is the range formula W = v₀²·sin(2α)/g, which holds for equal launch and landing heights. Insert the launch speed in m/s and the angle in degrees. Example: v₀ = 20 m/s, α = 45° gives W = 400 × sin(90°)/9.81 ≈ 40.8 m. Alternatively, work in two steps: first find the flight time from the vertical motion (t = 2·v₀·sinα/g), then substitute into the horizontal equation x = v₀·cosα·t. This route still works when launch and landing heights differ and the ready-made range formula fails.

In the range formula W = v₀²·sin(2α)/g the factor sin(2α) appears. The sine reaches its maximum of 1 exactly at 2α = 90°, i.e. at α = 45°. Flatter throws have plenty of horizontal speed but too little flight time; steeper throws fly long but barely move forward. 45° is the best compromise. Important: this only holds without air resistance and with equal launch and landing heights. When throwing from a height, as in shot put from shoulder level, the optimal angle is lower, typically 40° to 43°. Air resistance lowers it as well, for a football clearly below 45°.

At the highest point the vertical velocity is zero while the horizontal one continues unchanged. From v_y = v₀·sinα − g·t = 0 the rise time is t_s = v₀·sinα/g. Substituting into the y equation gives the closed formula H = v₀²·sin²α/(2g). Example: v₀ = 20 m/s, α = 60°: H = 400 × 0.75/19.62 ≈ 15.3 m, since sin²(60°) = 0.75. Make sure to form sin²α, first the sine, then the square. A common mistake is typing sin(α²). Alternatively, energy conservation also works: ½·v_y0² = g·H with the vertical launch speed v_y0.

The horizontal launch is the special case α = 0: the body starts without vertical initial velocity, usually from a height h, and the equations simplify to x = v₀·t and y = h − ½gt². The fall time then depends only on the height (t = √(2h/g)), exactly as in free fall, and the range is x = v₀·√(2h/g). In angled projectile motion the vertical launch component v₀·sinα is added, so the body first rises and then falls. Both cases follow the same principle: horizontal and vertical motions run independently and are calculated separately.

This is the superposition principle of kinematics: forces and accelerations act component-wise. Gravity points exactly downward and has no horizontal component. Therefore the horizontal velocity v₀·cosα stays constant during the whole flight, while the vertical motion is free fall with initial velocity v₀·sinα. Galileo confirmed this experimentally: a dropped ball and a horizontally launched ball hit the ground at the same time. The link between the two component motions lies solely in the shared time t. Eliminating t produces the parabola y(x), hence the name projectile parabola.