Projectile Motion Calculator

Calculate the trajectory, range, maximum height, and flight time for any projectile. This physics calculator visualizes the parabolic path and displays the equations of motion for your specific scenario.

What Is Projectile Motion?

Projectile motion is the motion of an object launched into the air that moves under the influence of gravity alone (neglecting air resistance). The path followed by a projectile is a parabola. Projectile motion combines constant horizontal velocity with uniformly accelerated vertical motion due to gravity.

Key Equations

The fundamental equations of projectile motion decompose the motion into horizontal (x) and vertical (y) components:

$$x(t) = v_0 \cos(\theta) \cdot t$$

$$y(t) = h_0 + v_0 \sin(\theta) \cdot t – \frac{1}{2}g t^2$$

where \(v_0\) is the initial speed, \(\theta\) is the launch angle, \(h_0\) is the initial height, and \(g\) is gravitational acceleration (9.81 m/s² on Earth).

Maximum Height

The projectile reaches its maximum height when the vertical velocity component equals zero. The time to reach peak height is:

$$t_{\text{peak}} = \frac{v_0 \sin(\theta)}{g}$$

The maximum height above the launch point is:

$$H = h_0 + \frac{v_0^2 \sin^2(\theta)}{2g}$$

Range and Flight Time

The total flight time is found by setting \(y(t) = 0\) and solving the resulting quadratic equation. For a projectile launched from ground level (\(h_0 = 0\)), the range simplifies to:

$$R = \frac{v_0^2 \sin(2\theta)}{g}$$

This equation shows that the maximum range occurs at a launch angle of 45°, and complementary angles (e.g., 30° and 60°) produce the same range.

Velocity Components

At any time \(t\) during flight, the velocity components are:

• Horizontal: \(v_x = v_0 \cos(\theta)\) (constant throughout flight)
• Vertical: \(v_y = v_0 \sin(\theta) – gt\) (decreases going up, increases going down)
• Speed: \(v = \sqrt{v_x^2 + v_y^2}\)

Trajectory Equation

Eliminating time from the parametric equations yields the trajectory equation relating \(y\) to \(x\) directly:

$$y = x \tan(\theta) – \frac{g x^2}{2 v_0^2 \cos^2(\theta)} + h_0$$

This confirms that the trajectory is a downward-opening parabola.

Effect of Launch Angle

The launch angle dramatically affects the trajectory. At 45°, range is maximized for flat ground. Steeper angles (closer to 90°) achieve greater height but shorter range. Shallower angles (closer to 0°) give a flatter trajectory with shorter flight time. In practice, factors like air resistance, spin, and wind make optimal angles different from the theoretical 45°.

Real-World Applications

Projectile motion governs sports (basketball arcs, golf drives, javelin throws), ballistics, fireworks trajectories, water fountain design, and even spacecraft launch trajectories (in the initial powered phase). Understanding projectile motion helps engineers design everything from catapults to Mars rovers’ landing approaches.

Projectile motion becomes manageable when you split one launch into two motions. Horizontal velocity stays constant in the ideal model, while gravity changes the vertical velocity.

How to use this calculator

Enter launch speed, launch angle, initial height, and gravitational acceleration using one unit system. The calculator resolves the initial velocity into \(v_x=v_0\cos\theta\) and \(v_y=v_0\sin\theta\).

Use 9.81 m/s² for a standard Earth estimate. Choose the angle from the horizontal, not the vertical, and keep the sign of the initial height consistent with the chosen ground level.

1. Resolve the launch velocity into horizontal and vertical components.
2. Use vertical motion to find flight time or the time at maximum height.
3. Multiply horizontal velocity by time to obtain range.
4. Check whether the equal-height assumptions behind shortcut formulas actually apply.

Worked example

Launch an object at 20 m/s and 30° from ground level. The components are about 17.32 m/s horizontally and 10 m/s vertically. Time to the top is \(10/9.81\), about 1.02 s, so total flight time is about 2.04 s.

The range is then about \(17.32\times2.04=35.3\) m. Maximum height above the launch point is \(10^2/(2\times9.81)\), about 5.10 m. These values assume no drag and equal launch and landing heights.

How to read the result

A 45° launch maximizes range only when launch and landing heights are equal and air resistance is ignored. Launching from a platform usually lowers the best angle because gravity already gives the projectile extra time in the air.

Complementary angles have the same ideal range at equal height because \(\sin(2\theta)\) matches for angles such as 30° and 60°. Their paths do not match: the steeper launch goes higher and stays airborne longer.

Common mistakes to avoid

• Using degrees when a software function expects radians.
• Applying the ground-to-ground range formula when the launch height is not zero.
• Treating speed as the horizontal component instead of resolving it with cosine.
• Expecting the ideal model to predict a light ball or high-speed projectile accurately in strong drag.

How to verify the result

At time zero, the computed position and velocity components must match the entered launch conditions. At the calculated landing time, substitute back into the vertical-position equation and confirm that the projectile reaches the chosen landing height.

Check that horizontal range equals horizontal velocity multiplied by flight time. At maximum height, vertical velocity should be zero. These independent relationships catch angle, sign, and degree-versus-radian errors.

Use consistent distance and time units and state the gravity value. The familiar 45-degree maximum-range rule applies only to equal launch and landing heights without air resistance, so do not use it as a universal check.

Limits of the calculation

The model ignores air resistance, wind, lift, spin, Earth’s curvature, and changing gravity. Those effects are small for many classroom problems but can dominate a baseball, golf ball, or long-range projectile.

Use numerical differential-equation methods when drag matters. The OpenStax treatment of projectile motion provides the standard constant-gravity model used here.

Related calculators

Use Kinematic Equations Calculator, Free Fall Calculator, Acceleration Calculator when the next part of the problem needs a different method.

Useful physics books and tools

For projectile motion, draw horizontal and vertical components and state whether air resistance is being ignored. Physics calculators work best after you draw the situation, choose a sign convention, and write every known value with its unit.

• Strengthen the model: Browse physics textbooks and problem books on Amazon. Choose a problem-focused book that explains diagrams, assumptions, and units.
• Handle the arithmetic: Browse scientific calculators on Amazon. A reliable scientific calculator helps with powers and trigonometry once the equation is set up correctly.

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