The Safety Parabola: The Boundary of Ideal Projectile Motion

What the safety parabola means

Launch many ideal projectiles from the same point with the same initial speed v₀ but with different elevation angles θ. Their individual paths form a family of parabolas. The upper boundary of every point that this family can reach is another parabola, called the safety parabola or parabola of safety.

A point below this boundary is generally reachable by two launch angles: a lower, flatter trajectory and a higher, steeper trajectory. A point on the boundary is reached by exactly one angle because the corresponding trajectory is tangent to the boundary. A point above the boundary cannot be reached with the selected initial speed in the ideal model.

The model assumes a level launch point, uniform gravitational acceleration g, no air resistance, and no curvature or rotation of Earth. The word safety is historical terminology; the curve is a mathematical reachability envelope, not a real-world safety guarantee.

Start from the laws of projectile motion

Choose Ox horizontally in the launch direction and Oy vertically upward. Resolve the initial velocity into v₀ₓ = v₀ cos θ and v₀ᵧ = v₀ sin θ. Horizontal acceleration is zero, while vertical acceleration is −g.

The time laws are x(t) = v₀ cos θ · t and y(t) = v₀ sin θ · t − gt²/2. Eliminating time with t = x/(v₀ cos θ) gives the trajectory equation for one chosen angle.

Derive the envelope step by step

Set u = tan θ and use 1/cos² θ = 1 + tan² θ. At a fixed horizontal coordinate x, the trajectory height becomes a quadratic function of u: y = xu − A(1 + u²), where A = gx²/(2v₀²).

The highest attainable y at that x occurs where the derivative with respect to u vanishes. Differentiating gives ∂y/∂u = x − (gx²/v₀²)u. Therefore the tangent trajectory has u = tan θ = v₀²/(gx).

Substitute this value of u back into the trajectory equation. The linear and quadratic terms combine to give yₛ = v₀²/(2g) − gx²/(2v₀²). This is the envelope of the complete family of trajectories.

Reachability from the quadratic discriminant

The same boundary follows by asking whether a target point (x, y) admits a real launch angle. Written as a quadratic in u = tan θ, the trajectory equation has zero, one, or two real solutions depending on its discriminant.

A positive discriminant gives two launch angles, a zero discriminant gives the single tangent angle, and a negative discriminant gives no real launch angle. Setting the discriminant equal to zero reproduces the safety-parabola equation.

Height, horizontal extent, and scale

At x = 0, the envelope reaches its vertical intercept Hₛ = v₀²/(2g). This is the limiting height approached by an increasingly vertical launch. Setting yₛ = 0 gives the positive ground intersection Dₛ = v₀²/g, which equals the maximum horizontal range obtained at 45°.

Both characteristic lengths are proportional to v₀². Doubling launch speed makes the safety height and safety distance four times larger. Increasing g compresses the entire reachable region.

How to read the calculator graph

The selected solid trajectory corresponds to one launch angle. The dashed safety parabola depends only on the fixed speed v₀ and g, so it does not move when only the angle changes. Comparison mode can retain several trajectories inside the same envelope.

The graph scales the axes independently to keep curves readable, so visual slope and curvature should not be measured directly from screen pixels. Use the numerical values for quantitative work and remember that air drag would make real trajectories shorter and non-parabolic.