Projectile Motion Calculator

Table of contents:
Do you want to analyze parabolic projectile motion? What about knowing more about what it means? Do you want to determine projectile motion equation values?  

Below, you can learn all this and more. Use a projectile motion calculator to learn about velocity, flight, and projectile ranges.

What is the Definition of Projectile Motion?

For an example of projectile motion, let us look at a golfer. Imagine him hitting the ball, blasting it forward and up. The further it moves toward the green, the slower its ascent becomes. It will then start its descent, showing promise for that elusive hole-in-one!

Pay attention to the movements that ball made. It would look like a curve (trajectory) in a parabolic shape. Anything forming that movement, like an archer shooting an arrow, is projectile motion.

In that motion, there is one force: gravity. If you involved a second force, then it would not be a projectile.

Analyzing Projectile Motion

Projectile motion might look complicated, but it involves logic. Once you know the initial velocity (v), launch angle (α), and initial height (h), use the calculator. The calculator uses the following steps to work out the remaining parameters for you.

1.	Calculate your velocity components.

- Horizontal velocity (Vx) = V x cos (α)
- Vertical velocity (Vy) = V x sin (α) 
- Three vectors (V, Vx, and Vy) = a right triangle

If the vertical velocity is zero, then you have horizontal projectile motion. If α = 90°, then it’s a freefall. 

2.	Establish the equations of motion.

- Horizontal distance traveled is x = Vx x t (time) 
- Vertical distance from the ground is y = h + Vy x t – g (gravity) x t² / 2

- Horizontal velocity = Vx 
- Vertical velocity = Vy – g x t

- Horizontal acceleration = 0 
- Vertical acceleration = -g (gravity acts on a projectile)

3.	Calculate the flight time
The flight time ends when the projectile hits the ground. You can determine this as being when the vertical distance to the ground is 0. When the height is 0, the formula is: 
Vy x t – g x t² / 2 = 0

Using that formula, you can establish the time of flight is: 

t = 2 x Vy / g = 2 x V x sin (α) / g

If you are adding elevation to the object, you solve a quadratic equation first. 

h + Vy x t – g x t² / 2 = 0

Solve that equation to get: 

t = [V x sin (α) + √(V x sin (α) ) ² + 2 x g x h ] / g

4.	Calculate the projectile’s range
The total horizontal distance during travel dictates the projectile’s range. If you launch a golf ball from the ground (height = 0), the formula will be: 

R = Vx x t = Vx x 2 x Vy / g

Simplify it to: 

R = V² x sin (2α) / g

What if the initial elevation is not 0? The long formula needs some minor alterations: 

R = Vx x t = V x cos(α) x [V x sin(α) + √ (V x sin(α)) ² + 2 x g x h)] / g

5.	Calculate the maximum height 
Your golfball will reach a point when it reaches its maximum altitude. When it does, it will start falling. The vertical velocity then changes from a positive number to a negative one. It becomes 0 for a moment in time. 

t (Vy = 0)

This equation: 
Vy – g x t (Vy = 0) = 0

Then becomes:

t (Vy = 0) = Vy / g

All you need to do then is find the vertical distance from the ground: 

hmax = Vy x t (vy=0) – g x (t(Vy=0))² / 2 = Vy² / (2 x g) = V² x sin(α)² / (2 x g)

When you launch an object from an initial height (h), you only need to add that value to the final formula: 

hmax = h + V² x sin(α)² / (2 x g)

Equations for Projectile Motion

After reading the above steps for calculating projectile motion, you might feel frazzled! You won’t remember them all, but these below are important: 

Launching an object from the ground (initial height h = 0)
1.	Horizontal velocity component 
Vx = V x cos (α)

2.	Vertical velocity component
Vy = V x sin (α)

3.	Time of flight 
t = 2 x Vy / g

4.	Projectile range 
R = 2 x Vx x Vy /g

5.	Maximum height 
hmax = Vy² / (2 x g)

Launching an object from an elevated position (initial height h > 0)
1.	Horizontal velocity component 
Vx = V x cos (α)

2.	Vertical velocity component 
Vy = V x sin (α)

3.	Flight time 
t = [Vy + √ (Vy² + 2 x g x h)] / g

4.	Projectile range 
R = Vx x [Vy + √ (Vy² + 2 x g x h)] / g

5.	Maximum height
hmax = h + Vy² / (2 x g)

Can you believe projectile motion calculations involve so much work? This projectile motion calculator will save you a lot of time! All you need are two values, and the calculator takes care of the rest.

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