Projectile Motion Simulator

What is projectile motion?

Projectile motion describes the path of an object launched into the air and subject only to gravity (ignoring air resistance). The object follows a parabolic trajectory — moving forward at constant horizontal velocity while simultaneously accelerating downward at 9.81 m/s squared. This combination of constant horizontal motion and accelerating vertical motion creates the characteristic curved path.

The key insight is that horizontal and vertical motions are independent. The horizontal velocity remains constant (no horizontal force without air resistance), while the vertical velocity changes due to gravity. You can analyze each direction separately using kinematics equations, then combine the results to get the complete trajectory.

Splitting initial velocity into components

The first step in any projectile problem is resolving the initial velocity v₀ into its horizontal and vertical components. If the launch angle is θ (measured from the horizontal), the components are: vₓ = v₀ · cos θ (horizontal) and vᵧ = v₀ · sin θ (vertical). The horizontal component vₓ stays constant throughout the flight because no horizontal force acts on the projectile (ignoring air resistance). The vertical component vᵧ decreases at the rate of g = 9.81 m/s² on Earth, reaching zero at the apex before increasing again on the way down.

This decomposition is the foundation of all projectile kinematics. Once you have vₓ and vᵧ, you can treat the horizontal direction as uniform motion (x = vₓ · t) and the vertical direction as uniformly accelerated motion (y = vᵧ · t − ½ g t²). Because these two equations share the same time variable t, you can link horizontal distance to vertical position and derive every quantity of interest.

Key projectile equations

• Range: R = (v0 squared * sin(2*theta)) / g — Maximum horizontal distance.
• Maximum height: H = (v0 squared * sin squared(theta)) / (2g).
• Time of flight: T = (2 * v0 * sin(theta)) / g.
• Optimal angle for maximum range: 45 degrees (without air resistance).

Worked example: 20 m/s at 30°

Suppose a ball is launched from ground level at v₀ = 20 m/s and θ = 30°. Here is the full solution step by step.

• Resolve components: vₓ = 20 · cos 30° ≈ 17.32 m/s; vᵧ = 20 · sin 30° = 10.00 m/s.
• Time of flight: T = 2 vᵧ / g = 2 × 10.00 / 9.81 ≈ 2.04 s.
• Maximum height: H = vᵧ² / (2g) = 10.00² / (2 × 9.81) ≈ 5.10 m.
• Horizontal range: R = vₓ × T = 17.32 × 2.04 ≈ 35.32 m. (Or using the direct formula: R = v₀² sin 2θ / g = 400 × sin 60° / 9.81 ≈ 35.31 m — the tiny difference is rounding.)

Try entering these values in the calculator above and you should see the same numbers. The trajectory diagram will show a relatively flat parabola; increase the angle to 45° and the range jumps to about 40.8 m with a higher arc, confirming that 45° gives the maximum range for a given initial speed.

Assumptions and limitations

This calculator, like most introductory physics tools, uses the idealized model of projectile motion. The assumptions are: (1) the projectile is a point mass; (2) gravity is constant and uniform (g = 9.81 m/s²); (3) there is no air resistance or drag; (4) the Earth’s curvature and rotation are negligible; (5) the launch and landing surfaces are flat. These assumptions hold reasonably well for slow, dense objects over short distances — for example, a shot put or a basketball thrown across a gym.

For high-speed or long-range projectiles, the model breaks down. A golf ball traveling at 70 m/s experiences drag that cuts its range by roughly 40% compared to the vacuum prediction. Artillery shells must account for the Coriolis effect due to Earth’s rotation. A space launch must treat gravity as decreasing with altitude and the trajectory becomes an orbital arc. Always choose your model based on the precision your application demands.

Common mistakes

• Degrees vs radians: most physics formulas use radians internally. When using a calculator or programming language, make sure to convert: θ(rad) = θ(°) × π / 180. Forgetting this conversion is the single most common source of wrong answers in projectile problems.
• Treating horizontal and vertical as the same: the horizontal velocity is constant; the vertical velocity is not. Never add vₓ and vᵧ directly when solving for time or height — keep the two directions separate until the very end.
• Using the wrong sign for g: take downward as negative. The vertical velocity equation is vᵧ(t) = vᵧ₀ − g t and position is y(t) = vᵧ₀ t − ½ g t². If you accidentally add g instead of subtracting, the projectile will accelerate upward forever.
• Ignoring launch height: the simple range formula R = v₀² sin 2θ / g only applies when the projectile lands at the same height it was launched. If there is a non-zero launch height (like a cliff), you must solve the quadratic y(t) = 0 to find the actual flight time before computing range.

How to use this tool

Enter the initial velocity and launch angle. The calculator shows the range, maximum height, time of flight, and the complete trajectory path. Adjust the angle to see how it affects the trajectory — notice that complementary angles (like 30 and 60 degrees) give the same range but different maximum heights.

Real-world considerations

In reality, air resistance significantly affects projectile motion, especially at high speeds. A baseball, football, or golf ball experiences drag that reduces range and changes the optimal launch angle to less than 45 degrees. The Magnus effect (spin-induced lift or curve) further complicates the trajectory. This calculator shows the ideal case without air resistance.

Frequently asked questions

Why is 45 degrees the optimal angle?

The range formula contains sin(2*theta), which is maximized when 2*theta = 90 degrees, meaning theta = 45 degrees. At this angle, the horizontal and vertical velocity components are equal, providing the best balance between going far and staying in the air long enough. With air resistance, the optimal angle drops to 35-42 degrees depending on the object.

Does the mass of the object affect projectile motion?

Without air resistance, no — all objects follow the same trajectory regardless of mass. This was famously demonstrated by Galileo (and later by astronaut David Scott on the Moon, dropping a hammer and feather). With air resistance, heavier objects are less affected by drag relative to their weight, so they travel farther.

What is the parabolic trajectory?

Eliminating time t from x = vₓ t and y = vᵧ t − ½ g t² gives y as a quadratic function of x: y = x tanθ − (g / 2 v₀² cos²θ) x². This is the equation of a parabola opening downward. The symmetry of the parabola means the projectile rises and falls along mirror-image paths, spending equal time ascending and descending (when launched and landing at the same height).

How does changing gravity affect the trajectory?

Lower gravity reduces the downward acceleration, so the projectile stays in the air longer and travels farther. On the Moon (g = 1.62 m/s²), the same 20 m/s at 30° launch gives a flight time of about 12.3 s and a range of roughly 214 m — about six times the Earth value. On Mars (g = 3.72 m/s²) the range would be approximately 93 m. The calculator lets you switch between Earth, Moon, Mars, or a custom gravity value to explore these differences interactively.