Projectile Motion Calculator
Calculate range, max height, and time of flight for projectile motion.
Understanding Projectile Motion Calculator
Projectile motion follows a parabolic trajectory under gravity (neglecting air resistance). Maximum range occurs at 45° launch angle.
$$R = \frac{v^2 \sin 2\theta}{g}$$
How to use this calculator
- Enter the launch speed, angle, and the value of gravity, usually \(9.8\,\text{m/s}^2\) on Earth.
- Choose the quantity you want to find, such as range, maximum height, or time of flight.
- Check that your angle is in degrees if the calculator expects degrees.
- Click calculate to get the result and the step-by-step working.
The formula explained
The range formula \(R = \frac{v^2 \sin 2\theta}{g}\) computes how far the projectile travels horizontally before landing. Here \(v\) is the launch speed, \(\theta\) is the launch angle, and \(g\) is the acceleration due to gravity.
- R = horizontal range, or total distance traveled along the ground
- v = launch speed
- \(\theta\) = launch angle above the horizontal
- g = acceleration due to gravity
- t = time of flight
- h = maximum height
Step by step method
- Find the launch speed \(v\), angle \(\theta\), and gravity \(g\).
- Use the range formula \(R = \frac{v^2 \sin 2\theta}{g}\) to find horizontal distance.
- For height and flight time, use the calculator's results or the standard projectile equations with the same inputs.
Worked example
Problem. A ball is launched at \(v = 20\,\text{m/s}\) with angle \(\theta = 30^\circ\) on Earth, where \(g = 9.8\,\text{m/s}^2\). Find the range.
- Substitute the values into the formula, \(R = \frac{20^2 \sin(2 \cdot 30^\circ)}{9.8}\).
- Simplify, \(R = \frac{400 \sin 60^\circ}{9.8}\). Since \(\sin 60^\circ \approx 0.866\), \(R \approx \frac{400 \cdot 0.866}{9.8}\).
- Compute the result, \(R \approx 35.3\,\text{m}\).
Answer. The range is about \(35.3\,\text{m}\).
Tips and common mistakes
- Make sure the angle matches the calculator setting, because \(30^\circ\) is not the same as \(30\) radians.
- The formula assumes the projectile lands at the same height it was launched from, unless the calculator states otherwise.
Frequently asked questions
How do I use the projectile motion calculator?+
Enter the launch speed, angle, and the value of gravity if needed, then the calculator returns the range, maximum height, and time of flight. It uses the standard projectile equations and shows the results step by step.
What does the range formula R = v^2 sin(2θ) / g mean?+
This formula gives the horizontal distance traveled when a projectile lands at the same height it was launched from, assuming no air resistance. The range depends on the square of the launch speed, the launch angle through sin(2θ), and inversely on gravity.
Why does the calculator give the same range for angles like 30° and 60°?+
Because sin(2θ) is the same for complementary angles, so 30° and 60° both give sin(60°). That means, with the same speed and gravity, they produce the same range even though the height and flight time are different.
What happens if the projectile is launched from a height above the landing point?+
The simple range formula no longer applies exactly, because it assumes launch and landing happen at the same height. In that case, the time of flight and range must be found using the vertical motion equation, and the calculator may need a more general setup if it supports it.
What is the difference between range, maximum height, and time of flight?+
Range is the horizontal distance traveled, maximum height is the highest vertical point reached, and time of flight is how long the projectile stays in the air. They describe different parts of the same motion, so a change in angle or speed can affect each one differently.
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