Projectile Motion Calculator
Calculate projectile motion using the fundamental kinematic equations. Find range, maximum height, time of flight, and impact velocity from initial velocity, launch angle, and launch height. Perfect for physics students, engineers, and anyone working with ballistic trajectories.
What Is Projectile Motion?
Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. It is a form of two-dimensional kinematics where the horizontal and vertical components of motion are independent of each other.
In ideal projectile motion (neglecting air resistance), the horizontal velocity remains constant, while the vertical velocity changes uniformly due to gravity. The resulting path is a parabola.
This calculator supports finding:
- Range (R): The horizontal distance traveled by the projectile.
- Maximum Height (H): The highest vertical position reached.
- Time of Flight (T): The total time the projectile stays in the air.
- Impact Velocity: The speed and angle of the projectile when it lands.
Projectile Motion Formulas
The calculator uses the following kinematic equations:
Horizontal Velocity: v₀ₓ = v₀ × cos(θ)
Vertical Velocity: v₀ᵧ = v₀ × sin(θ)
Time of Flight: T = (v₀ᵧ + √(v₀ᵧ² + 2 × g × h₀)) / g
Maximum Height: H = h₀ + v₀ᵧ² / (2 × g)
Range: R = v₀ₓ × T
Impact Velocity: v = √(v₀ₓ² + (v₀ᵧ − g × T)²)
Where v₀ is the initial velocity, θ is the launch angle, g is gravitational acceleration, and h₀ is the initial launch height.
Why Use This Projectile Motion Calculator?
- Complete Results: Calculates range, max height, time of flight, and impact velocity simultaneously.
- Trajectory Chart: Visualizes the parabolic path of the projectile.
- Component Breakdown: Shows horizontal and vertical velocity components and time to max height.
- Unit Flexibility: Supports both metric and imperial units.
- Formula Display: See the formulas used for each calculation.
- Free & Private: No registration, no data storage.
❓ Projectile Motion FAQ
What is projectile motion?
Projectile motion is the motion of an object thrown or projected into the air, subject only to gravity. It follows a parabolic path and can be analyzed by separating horizontal and vertical motion.
What is the formula for range?
For a projectile launched from ground level: R = v₀² sin(2θ) / g. For launch from a height: R = v₀ₓ × T, where T is the total time of flight.
What is the formula for maximum height?
Maximum height is H = h₀ + v₀ᵧ² / (2g), where v₀ᵧ = v₀ sin(θ) is the initial vertical velocity and h₀ is the launch height.
What is the formula for time of flight?
Time of flight is T = (v₀ᵧ + √(v₀ᵧ² + 2gh₀)) / g. For ground-level launch (h₀ = 0), this simplifies to T = 2v₀ sin(θ) / g.
Why is 45° the optimal launch angle for maximum range?
At 45°, sin(2θ) = sin(90°) = 1, which is the maximum value. This makes R = v₀² / g, the maximum possible range for a given initial velocity from ground level.
What is the difference between horizontal and vertical motion?
Horizontal motion has constant velocity (no acceleration in the x-direction). Vertical motion has constant acceleration due to gravity (a = -g in the y-direction).
How does initial height affect the projectile?
Launching from a height increases both the time of flight and the range. The projectile has more time to travel horizontally before hitting the ground.
What is impact velocity?
Impact velocity is the speed of the projectile when it lands. It is calculated as v = √(v₀ₓ² + (v₀ᵧ − gT)²), where T is the time of flight.
Does air resistance affect projectile motion?
Yes, in real-world scenarios, air resistance reduces range and maximum height. This calculator assumes ideal conditions (no air resistance) for simplicity.
Can I use this calculator for any projectile?
Yes, as long as air resistance is negligible. It works for balls, bullets, arrows, and any object launched at an angle.
What units can I use?
This calculator supports both metric (meters, m/s, m/s²) and imperial (feet, ft/s, ft/s²) units. Select your preferred system from the dropdown.
What is the acceleration due to gravity?
On Earth, g = 9.81 m/s² (32.2 ft/s²). You can change this value for other planets or scenarios (e.g., Moon: 1.62 m/s², Mars: 3.71 m/s²).