# Projectile Motion Calculator

Our projectile motion calculator is a tool that helps you analyze the parabolic projectile motion. It can find the time of flight, but also the components of velocity, the range of the projectile and maximum height of flight. Continue reading to get familiar with the projectile motion definition and to determine the abovementioned values, using the projectile motion equations.

## Projectile motion definition

Imagine an archer sending an arrow in the air. It starts moving up and forward, at some inclination to the ground. The further it flies, the slower its ascend is – and finally, it starts descending, moving now downwards and forwards and finally hitting the ground again. If you could trace its path, it would be a curve – a parabola. Any object moving in such a way is in projectile motion.

Only one force acts on a projectile – the gravity force. Air resistance is always omitted.

## Projectile motion analysis

Projectile motion is pretty logical. Let’s assume you know the initial velocity of the object `v`

and the angle of launch `θ`

. Our projectile motion calculator follows these steps to find all remaining parameters:

**Calculate the components of velocity.**

- The horizontal velocity component
`vx`

is equal to`v * cos(θ)`

. - The vertical velocity component
`vy`

is equal to`v * sin(θ)`

. - Three vectors -
`v`

,`vx`

and`vy`

- form a right triangle.

**Write down the equations of motion.**

- Horizontal distance traveled can be expressed as
`x = vx * t`

, where`t`

is the time. - Vertical distance from the ground is described by the formula
`y = vy * t – g * t² / 2`

, where`g`

is the gravity acceleration.

- Horizontal velocity is equal to
`vx`

. - Vertical velocity can be expressed as
`vy – g * t`

.

- Horizontal acceleration is equal to 0.
- Vertical acceleration is equal to
`-g`

(because only gravity acts on the projectile).

**Calculate the time of flight.**

- Flight ends when the projectile hits the ground. We can say that it happens when the vertical distance from the ground is equal to 0, or
`vy * t – g * t² / 2 = 0`

. - From that equation, we find that the time of flight is
`t = 2 * vy / g`

.

**Calculate the range of the projectile.**

- The range of the projectile is the total horizontal distance traveled in the flight time.
- We can write it down as
`R = vx * t = vx * 2 * vy / g`

.

**Calculate the maximum height.**

- When the projectile reaches the maximum height, is stops moving up and starts falling. It means that its vertical velocity component changes from positive to negative – in other words, it is equal to 0 for a brief moment at time
`t(vy=0)`

. - If
`vy – g * t(vy=0) = 0`

, then we can reformulate this equation to`t(vy=0) = vy / g`

. - Now, we simply find the vertical distance from the ground at that time:
`ymax = vy * t(vy=0) – g * (t(vy=0))² / 2 = vy² / (2 * g)`

.

## Projectile motion equations

Uff, that was a lot of calculations! Let's sum that up to form the most important projectile motion equations:

- Horizontal velocity component:
`vx = v * cos(θ)`

- Vertical velocity component:
`vy = v * sin(θ)`

- Time of flight:
`t = 2 * vy / g`

- Range of the projectile:
`R = 2 * vx * vy / g`

- Maximum height:
`ymax = vy² / (2 * g)`