Calculate Position of A Projectile with Distance

This calculator determines the vertical position of a projectile at any given horizontal distance from the launch point. It's useful for physics students, engineers, and anyone analyzing projectile motion.

How to Calculate Projectile Position

Projectile motion occurs when an object is launched into the air and moves only under the acceleration of gravity. The position of the projectile at any given distance can be calculated using the following steps:

Determine the initial velocity (v₀) and launch angle (θ) of the projectile.
Calculate the time of flight (t) for the projectile to reach the desired distance.
Use the time to find the vertical position (y) at that distance.

The key assumptions are:

Air resistance is negligible
Gravity is constant (9.81 m/s²)
Projectile is launched from ground level

Note: This calculator assumes standard Earth gravity. For other celestial bodies, you would need to adjust the gravitational constant accordingly.

Projectile Motion Formula

The vertical position (y) of a projectile at horizontal distance (x) is given by:

y = (v₀ sinθ)t - (1/2)gt²

where:

y = vertical position (meters)
v₀ = initial velocity (meters/second)
θ = launch angle (degrees)
t = time (seconds)
g = acceleration due to gravity (9.81 m/s²)

The time to reach distance x is calculated from:

t = x / (v₀ cosθ)

Combining these gives the complete position formula:

y = (v₀ sinθ)(x / (v₀ cosθ)) - (1/2)g(x / (v₀ cosθ))²

Simplified to:

y = x tanθ - (g x²) / (2 v₀² cos²θ)

Worked Example

Let's calculate the position of a projectile 10 meters from the launch point with these parameters:

Initial velocity (v₀) = 20 m/s
Launch angle (θ) = 45°
Distance (x) = 10 m

Convert angle to radians: 45° = 0.785 radians
Calculate time: t = 10 / (20 * cos(0.785)) = 10 / (20 * 0.707) ≈ 0.707 seconds
Calculate vertical position: y = (20 * sin(0.785)) * 0.707 - (1/2) * 9.81 * (0.707)² ≈ 7.07 - 2.5 ≈ 4.57 meters

The projectile is approximately 4.57 meters above the ground when it's 10 meters horizontally from the launch point.

Projectile Position at Different Distances
Distance (m)	Height (m)
5	5.00
10	4.57
15	3.25
20	1.00

FAQ

What if the projectile lands before reaching the desired distance?: The calculator will show a negative position, indicating the projectile has already landed. You would need to choose a distance within the projectile's range.
Can this calculator handle different units?: Currently, the calculator uses meters and seconds. For other units, you would need to convert them to the required units before input.
What if the launch angle is 90 degrees?: At 90 degrees, the projectile moves purely vertically. The horizontal distance will always be zero, and the position formula simplifies to the standard vertical motion equation.
How accurate are the calculations?: The calculations are based on standard projectile motion physics equations and assume ideal conditions with no air resistance. Real-world results may vary.