Calculate projectile motion with this free online calculator. Find flight time, maximum height, horizontal range, and impact speed using initial velocity, launch angle, launch height, and gravity. Ideal for physics students, homework, and motion experiments.
Projectile Motion Results
Horizontal velocity: Vx = V × cos(θ)
Initial vertical velocity: Vy = V × sin(θ)
Vertical position: y = h + Vy × t − ½gt²
Flight time: T = [Vy + √(Vy² + 2gh)] / g
Maximum height: H = h + Vy² / (2g)
Horizontal range: R = Vx × T
Impact vertical velocity: Vy − gT
Impact speed: √[Vx² + (Vy − gT)²]
V = initial speed, θ = launch angle, h = initial height, g = gravitational acceleration, and T = flight time. These formulas assume constant gravity and no air resistance.How to Calculate Projectile Motion
Projectile motion describes the movement of an object launched into the air under the influence of gravity. Examples include throwing a ball, kicking a football, or launching an object from an elevated platform.
A projectile's motion has two components: horizontal motion and vertical motion. In the ideal model, horizontal velocity remains constant, while gravity changes the vertical velocity throughout the flight.
This Projectile Motion Calculator uses the initial speed, launch angle, initial height, and gravitational acceleration to calculate the object's trajectory-related measurements.
Projectile Motion Formulas
The main equations used in projectile motion are:
Horizontal velocity
Vx = V × cos(θ)
Initial vertical velocity
Vy = V × sin(θ)
Flight time
T = [Vy + √(Vy² + 2gh)] / g
Maximum height above ground
H = h + Vy² / (2g)
Horizontal range
R = Vx × T
Impact speed
Impact speed = √[Vx² + (Vy − gT)²]
Where:
- V = Initial speed in meters per second
- θ = Launch angle measured from the horizontal
- Vx = Horizontal velocity component
- Vy = Initial vertical velocity component
- h = Initial height above the landing ground
- g = Gravitational acceleration
- T = Total flight time
- H = Maximum height above the ground
- R = Horizontal distance traveled
These equations assume constant gravitational acceleration and negligible air resistance.
Example of Projectile Motion Calculation
Suppose a ball is thrown from a platform 10 meters above the ground with an initial speed of 20 m/s at an angle of 45°. Assume gravity is 9.81 m/s².
Step 1: Calculate the velocity components
Horizontal velocity:
Vx = 20 × cos(45°)
Vx ≈ 14.14 m/s
Vertical velocity:
Vy = 20 × sin(45°)
Vy ≈ 14.14 m/s
Step 2: Calculate the flight time
T = [14.14 + √(14.14² + 2 × 9.81 × 10)] / 9.81
T ≈ 3.47 seconds
Step 3: Calculate the maximum height
H = 10 + 14.14² / (2 × 9.81)
H ≈ 20.19 meters
Step 4: Calculate the horizontal range
R = 14.14 × 3.47
R ≈ 49.09 meters
Step 5: Calculate the impact speed
The impact speed accounts for the increase in downward velocity as the ball falls.
Impact speed = √(20² + 2 × 9.81 × 10)
Impact speed ≈ 24.42 m/s
Therefore, the ball stays in the air for approximately 3.47 seconds, reaches a maximum height of 20.19 meters above the ground, travels about 49.09 meters horizontally, and hits the ground at approximately 24.42 m/s.
Why Does Initial Height Matter?
The initial height determines how far the projectile must fall before reaching the ground. An object launched from a higher position generally stays in the air longer than an object launched with the same speed and angle from ground level.
For example, a ball thrown horizontally from a balcony can travel a horizontal distance before hitting the ground, even though its initial vertical velocity is zero.
This is why including the initial height makes a projectile motion calculator useful for a wider range of physics problems.
How Does Launch Angle Affect Range?
The launch angle determines how the initial speed is divided between horizontal and vertical motion.
- At a low angle: More speed is directed horizontally, and the trajectory stays relatively low.
- At a high angle: More speed is directed upward, increasing the maximum height.
- At 45 degrees: A projectile launched and landing at the same height achieves its maximum horizontal range at this angle under ideal conditions.
When the launch and landing heights differ, the angle that gives the maximum range may also differ.
Frequently Asked Questions
1. What is projectile motion?
Projectile motion is the movement of an object through the air under gravity, assuming air resistance is negligible.
2. Can I calculate projectile motion from an elevated position?
Yes. Enter the initial height above the landing ground. The calculator uses that height to determine flight time, maximum height, range, and impact speed.
3. What is the difference between launch speed and impact speed?
Launch speed is the object's speed when it begins moving. Impact speed is its speed immediately before reaching the ground. When an object lands below its launch point, its impact speed is generally higher if air resistance is ignored.
4. What is the ideal launch angle for maximum range?
For an object launched and landing at the same height, 45 degrees gives the maximum horizontal range for a fixed initial speed in the ideal model. Different launch and landing heights can change the best angle.
5. Does projectile motion include air resistance?
This calculator does not include air resistance, wind, or aerodynamic lift. Real-world trajectories can differ from the calculated results.
6. What is the difference between horizontal and vertical velocity?
Horizontal velocity describes movement parallel to the ground, while vertical velocity describes upward or downward movement. Gravity changes vertical velocity but does not change horizontal velocity in the ideal model.