🎯 Projectile Motion Calculator
Calculate projectile motion
How to Use This Calculator
Enter Initial Velocity
Input the initial speed at which the projectile is launched, measured in meters per second (m/s). This is the magnitude of the velocity vector at launch.
Enter Launch Angle
Input the angle at which the projectile is launched above the horizontal, measured in degrees. For maximum range (without air resistance), use 45°.
Enter Initial Height (Optional)
If the projectile starts from a height above ground level, enter that initial height in meters. Leave as 0 if launched from ground level.
Set Gravity (Optional)
The default gravity is 9.81 m/s² (Earth). You can change this for calculations on other planets or in different gravitational fields.
Click Calculate
Press the "Calculate" button to compute range, maximum height, time of flight, time to reach maximum height, and final velocity.
Formula
Range: R = v₀x × t_flight
Max Height: H = h₀ + (v₀² sin²(θ)) / (2g)
Time of Flight: T = (v₀ sin(θ) + √(v₀² sin²(θ) + 2gh₀)) / g
Where:
vâ‚€ = Initial velocity (m/s)
θ = Launch angle (degrees)
hâ‚€ = Initial height (m)
g = Gravitational acceleration (m/s²)
v₀x = v₀ cos(θ) - horizontal component
v₀y = v₀ sin(θ) - vertical component
Example 1: Projectile from ground level at 45°
Given: v₀ = 30 m/s, θ = 45°, h₀ = 0 m, g = 9.81 m/s²
v₀x = 30 × cos(45°) = 21.21 m/s
v₀y = 30 × sin(45°) = 21.21 m/s
Time of flight: T = (21.21 + √(21.21² + 0)) / 9.81 = 4.32 s
Range: R = 21.21 × 4.32 = 91.63 m
Max height: H = 0 + (21.21²) / (2 × 9.81) = 22.94 m
Example 2: Projectile from elevated position
Given: v₀ = 25 m/s, θ = 60°, h₀ = 10 m, g = 9.81 m/s²
v₀y = 25 × sin(60°) = 21.65 m/s
T = (21.65 + √(21.65² + 2×9.81×10)) / 9.81 = 5.18 s
R = 25 × cos(60°) × 5.18 = 64.75 m
H = 10 + (21.65²) / (2 × 9.81) = 33.87 m
About Projectile Motion Calculator
The Projectile Motion Calculator is a comprehensive physics tool for analyzing the motion of objects launched into the air at an angle. Projectile motion is a form of motion where an object moves in a parabolic path under the influence of gravity, with no air resistance. This calculator computes all key parameters including horizontal range, maximum height, total time of flight, time to reach maximum height, and final velocity. It's essential for physics students, engineers, sports scientists, and anyone working with ballistics or trajectory analysis.
When to Use This Calculator
- Physics Homework: Solve projectile motion problems involving range, height, and time calculations
- Sports Analysis: Analyze trajectories of balls in sports like baseball, basketball, football, or golf
- Engineering Design: Calculate trajectories for projectiles in mechanical systems or safety assessments
- Ballistics: Analyze the motion of bullets, arrows, or other projectiles
- Educational Purposes: Understand the relationship between launch angle, velocity, and trajectory
- Safety Planning: Calculate landing distances for fireworks, rockets, or other launched objects
Why Use Our Calculator?
- ✅ Comprehensive Results: Calculates range, max height, time of flight, and final velocity in one calculation
- ✅ Handles Initial Height: Accounts for projectiles launched from elevated positions
- ✅ Accurate Formulas: Uses precise kinematic equations for projectile motion
- ✅ Customizable Gravity: Adjust gravitational acceleration for different planets or scenarios
- ✅ Educational Value: Shows formulas and helps understand trajectory physics
- ✅ Multiple Outputs: Provides all key trajectory parameters simultaneously
Common Applications
Sports Physics: Analyze the trajectory of a basketball shot, football kick, or baseball hit to understand optimal launch angles and velocities for maximum range or accuracy.
Engineering Projects: Calculate trajectories for projectiles in mechanical launching systems, automated throwing devices, or safety barrier assessments.
Physics Education: Help students understand how launch angle, initial velocity, and initial height affect trajectory, demonstrating key concepts in kinematics and two-dimensional motion.
Safety Planning: Calculate landing distances and maximum heights for fireworks, rockets, or other projectiles to ensure safe launch distances and clearances.
Tips for Best Results
- Optimal Angle: For maximum range (without initial height), use 45° launch angle
- Use Consistent Units: Ensure all inputs use meters (m) for distance and m/s for velocity
- Consider Air Resistance: This calculator assumes no air resistance; real-world trajectories will differ
- Initial Height Matters: Don't forget to include initial height if the projectile starts elevated
- Angle Range: Launch angles should be between 0° (horizontal) and 90° (vertical) for meaningful results
Frequently Asked Questions
What is the optimal launch angle for maximum range?
For projectiles launched from ground level (h₀ = 0), the optimal launch angle is 45° for maximum range. This gives equal horizontal and vertical velocity components. For elevated launches, the optimal angle is slightly less than 45°.
How does initial height affect the trajectory?
Initial height increases both the maximum height and time of flight. The range also increases because the projectile has more time in the air. The formula accounts for initial height in the time of flight calculation.
What happens if I change the launch angle?
Launch angle affects all trajectory parameters. At 45° (ground level), range is maximum. At 90° (vertical), range is zero but height is maximum. At 0° (horizontal), range exists but height is minimal (just initial height).
Does this calculator account for air resistance?
No, this calculator assumes no air resistance, which is valid for low speeds and dense objects. For high-speed projectiles or light objects, air resistance significantly affects trajectory, reducing range and maximum height.
Can I use this for projectiles on other planets?
Yes! Simply change the gravity value. For example, use g = 1.62 m/s² for the Moon, g = 3.71 m/s² for Mars, or g = 24.79 m/s² for Jupiter. All calculations will adjust accordingly.
What is the relationship between time of flight and maximum height time?
The time to reach maximum height is exactly half the total time of flight (for symmetric trajectories with hâ‚€ = 0). This is because the upward and downward portions of the trajectory are symmetric when launched from ground level.