Projectile Physics Calculator: Solve Range, Height & Time

This projectile motion calculator works out the key flight values for an object launched at an angle with an initial velocity: maximum height, time of flight, and horizontal range. You enter the launch speed, launch angle, and optionally the starting height, and it returns the trajectory results using standard physics equations. It assumes ideal projectile motion with no air resistance, which is the standard model used in introductory physics.

Formula

The calculator uses the standard kinematic equations for constant acceleration due to gravity. The three main outputs are:

• Horizontal range (R): R = (v·cosθ / g) · (v·sinθ + √((v·sinθ)² + 2·g·h))
• Maximum height (H): H = h + (v·sinθ)² / (2·g)
• Time of flight (T): T = (v·sinθ + √((v·sinθ)² + 2·g·h)) / g

Where:

• v = initial velocity (m/s)
• θ = launch angle (degrees)
• h = initial height above ground (m)
• g = gravitational acceleration = 9.80665 m/s²

If the initial height is zero, the formulas simplify to the familiar R = v²·sin(2θ)/g, H = v²·sin²θ/(2g), and T = 2·v·sinθ/g.

Worked Example

Using the default values: initial velocity = 20 m/s, angle = 45°, initial height = 0 m.

1. Break the velocity into components: vx = 20·cos45° = 14.142 m/s, vy = 20·sin45° = 14.142 m/s.
2. Time to max height: t_up = vy / g = 14.142 / 9.80665 = 1.442 s.
3. Max height: H = 0 + (14.142)² / (2·9.80665) = 200 / 19.6133 = 10.19 m.
4. Total time of flight: T = 2 × 1.442 = 2.884 s.
5. Range: R = vx × T = 14.142 × 2.884 = 40.79 m.

The calculator gives 40.79 m for range, 10.19 m for max height, and 2.88 s for time of flight, matching the hand calculation.

When to Use This Calculator

This is the right tool for physics homework, exam practice, or any basic trajectory problem where air resistance is ignored. It is commonly used for:

• Solving textbook projectile problems (cannonballs, balls thrown at an angle, etc.)
• Checking answers for lab reports or assignments
• Understanding how launch angle affects range and height
• Designing simple experiments with projectiles (e.g., water rockets, model catapults)

It is not suitable for real-world objects like golf balls, bullets, or rockets, where air drag is significant. For those, you would need a numerical simulation that includes drag coefficients and air density.

How to Read the Results

• Range: the horizontal distance from the launch point to where the projectile lands on the same level (or the level specified by initial height).
• Max height: the highest point above the ground (or above launch point if initial height is zero).
• Time of flight: the total time from launch until it hits the ground.

Note that the range is maximized at a 45° launch angle when the landing height equals the launch height. If the landing height is lower (e.g., launching from a cliff), the optimal angle is slightly less than 45°.

Common Mistakes and Misconceptions

• Using degrees instead of radians: Most calculators expect degrees for trig functions. This calculator uses degrees, but if you use a different tool, check the mode.
• Forgetting initial height: If the projectile starts above ground, the time of flight increases, and so does the range. Many simple formulas assume h=0.
• Ignoring air resistance: The real world has drag. For a baseball thrown at 20 m/s, air resistance can reduce range by 20-30%.
• Mixing units: Always use meters and seconds. If you have km/h, convert to m/s by dividing by 3.6.

Limits of the Formula

The equations assume:

• No air resistance (vacuum)
• Constant gravitational acceleration (9.80665 m/s²)
• Flat, non-rotating Earth (no Coriolis effect)
• Projectile is a point mass (no spin, no lift)

For typical school problems, these assumptions are fine. For real projectiles, the results are approximate. The error grows with speed and altitude. For a 100 m/s projectile, the range can be off by more than 50% due to drag.

Frequently Asked Questions

What is the optimal angle for maximum range?

When the launch and landing heights are the same, 45° gives the maximum range. If the landing point is lower (e.g., throwing from a cliff), the optimal angle is less than 45°, typically around 40-42° depending on the height difference. If the landing point is higher, the optimal angle is greater than 45°.

Does mass affect projectile motion?

In ideal projectile motion (no air resistance), mass does not affect the trajectory. A feather and a cannonball would follow the same path if launched with the same velocity and angle. In reality, air resistance makes mass matter: heavier objects with the same shape are less affected by drag.

How do I convert from km/h to m/s?

Divide by 3.6. For example, 72 km/h ÷ 3.6 = 20 m/s. This is a common step because many real-world speeds are given in km/h, but the formulas require m/s.

What if I only know the range and angle?

You can rearrange the range formula to solve for initial velocity: v = √(g·R / sin(2θ)). For example, if R=40 m and θ=45°, then v = √(9.80665·40 / sin(90°)) = √392.266 = 19.8 m/s. This is useful for reverse-engineering problems.

Why does the calculator give a different answer than my textbook?

Check if your textbook uses g = 9.8 m/s² instead of 9.80665, or if it rounds intermediate steps. Also check whether the book assumes the projectile lands at the same height or at a different height. Small rounding differences are normal.

Can I use this for a projectile launched straight up?

Yes. Set the angle to 90°. The range will be 0 m, the max height will be v²/(2g), and the time of flight will be 2v/g (if starting from ground level). This is a special case of the same formulas.

Related

• 24Na Beta Decay Calculator: Physics & Half-Life
• A Level Physics Calculator: What to Use & Why
• AP Physics Score Calculator: How It Works

in v | Initial velocity | m/s | 20 | 0
in a | Launch angle | degrees | 45 | 0
in h | Initial height | m | 0 | 0
out range | Horizontal range | m | (v * cosd(a) / 9.80665) * (v * sind(a) + sqrt((v * sind(a))^2 + 2 * 9.80665 * h)) | 2
out height | Max height | m | h + (v * sind(a))^2 / (2 * 9.80665) | 2
out time | Time of flight | s | (v * sind(a) + sqrt((v * sind(a))^2 + 2 * 9.80665 * h)) / 9.80665 | 2