Projectile Motion Calculator
Calculate projectile trajectory, range, maximum height, and flight time. Solve physics problems involving objects launched at angles with initial velocity.
Projectile Motion Formulas
Position
x(t) = v₀·cos(θ)·t
y(t) = h₀ + v₀·sin(θ)·t - ½gt²
Velocity
vₓ = v₀·cos(θ) (constant)
vᵧ(t) = v₀·sin(θ) - gt
Key Results
Max height: H = (v₀·sin(θ))² / 2g
Range: R = v₀²·sin(2θ) / g
Notes
g = 9.81 m/s² (Earth gravity)
Max range at 45° (flat ground)
Related Calculators
About This Calculator
Projectile motion describes the path of any object thrown or launched into the air, subject only to gravity and air resistance. From a baseball to a cannonball, from a basketball shot to a rocket trajectory, understanding projectile motion is fundamental to physics, engineering, and sports science.
What is Projectile Motion? When an object is launched with an initial velocity at an angle, it follows a curved path called a parabola. The horizontal motion is constant (no acceleration), while the vertical motion is affected by gravity, causing the classic arc we see in thrown objects.
Why It Matters:
- Essential for physics education and problem-solving
- Used in ballistics and military applications
- Critical for sports science and coaching
- Important in video game physics and simulations
- Applied in aerospace and rocket science
Key Components:
- Initial Velocity (v₀): Launch speed
- Launch Angle (θ): Angle above horizontal
- Gravity (g): Downward acceleration (9.81 m/s²)
- Range: Horizontal distance traveled
- Maximum Height: Peak of the trajectory
This calculator handles trajectory analysis, angle finding, and velocity calculations. For related physics, see our Velocity Calculator and Free Fall Calculator.
How to Use the Projectile Motion Calculator
- 1Select your calculation type based on what you know and want to find.
- 2For trajectory: Enter initial velocity and launch angle.
- 3Optionally enter launch height if not launching from ground level.
- 4For finding angle: Enter velocity and target range.
- 5For finding velocity: Enter angle and target range.
- 6Review the calculated trajectory parameters.
- 7Examine the trajectory data table for position over time.
- 8Note that 45° gives maximum range on flat ground.
- 9Two angles often reach the same range (complementary paths).
- 10Air resistance is neglected in these calculations.
The Physics of Projectile Motion
Projectile motion combines two independent motions.
The Two Components
Horizontal Motion (x-direction):
- No acceleration (ignoring air resistance)
- Constant velocity: vₓ = v₀·cos(θ)
- Position: x(t) = v₀·cos(θ)·t
Vertical Motion (y-direction):
- Constant acceleration: g = 9.81 m/s² downward
- Velocity changes: vᵧ(t) = v₀·sin(θ) - gt
- Position: y(t) = h₀ + v₀·sin(θ)·t - ½gt²
The Parabolic Path
Eliminating time from the equations:
y = x·tan(θ) - (g·x²)/(2v₀²·cos²θ)
This is a parabola opening downward.
Independence of Motions
Key insight: Horizontal and vertical motions are independent.
- A bullet fired horizontally and one dropped simultaneously hit the ground at the same time
- The horizontal velocity doesn't affect falling rate
- This principle is fundamental to physics
Key Trajectory Parameters
Important results that can be calculated from initial conditions.
Maximum Height
When vertical velocity becomes zero:
H = (v₀·sin(θ))² / (2g)
Time to reach maximum height: t_max = v₀·sin(θ) / g
Total Flight Time
For launch and landing at same height:
T = 2·v₀·sin(θ) / g
For different heights, solve the quadratic equation for y = 0.
Horizontal Range
For flat ground:
R = v₀²·sin(2θ) / g
Maximum range occurs at θ = 45°: R_max = v₀² / g
Two Angles, Same Range
For any range less than R_max, two angles work:
- Low trajectory: θ
- High trajectory: 90° - θ
Example: 30° and 60° give the same range
- 30°: lower, faster flight
- 60°: higher, longer flight time
Common Scenarios and Examples
Applying projectile motion to real situations.
Example 1: Sports - Football Punt
Given: v₀ = 25 m/s, θ = 50°
Calculate:
- vₓ = 25·cos(50°) = 16.1 m/s
- vᵧ = 25·sin(50°) = 19.2 m/s
- Max height: H = 19.2²/(2×9.81) = 18.8 m
- Flight time: T = 2×19.2/9.81 = 3.91 s
- Range: R = 16.1 × 3.91 = 62.9 m
Example 2: Cliff Jump
Given: v₀ = 5 m/s, θ = 30°, h₀ = 10 m
Calculate:
- vₓ = 5·cos(30°) = 4.33 m/s
- vᵧ = 5·sin(30°) = 2.5 m/s
- Using y = 0: 0 = 10 + 2.5t - 4.9t²
- Flight time: T = 1.7 s
- Range: R = 4.33 × 1.7 = 7.4 m
Example 3: Basketball Shot
Given: Range = 6 m, hoop 3 m high, release at 2 m
Strategy:
- Need to clear the hoop: Δh = 1 m
- Optimal entry angle: 45-52°
- Work backward to find required v₀ and θ
Example 4: Cannon Problem
Goal: Hit target 200 m away
Solution: At θ = 45° (max range):
- v₀ = √(g·R) = √(9.81×200) = 44.3 m/s
At θ = 30°:
- v₀ = √(g·R/sin(60°)) = 47.6 m/s
Effects of Launch Height
Starting above or below the landing point changes the analysis.
Launching from Height
When h₀ > 0:
- Flight time increases
- Range increases
- Optimal angle decreases (less than 45°)
Modified range equation: More complex - requires solving the full trajectory equation.
Approximate Effect
For small heights relative to range:
- Optimal angle ≈ 45° - arctan(h/R)/2
Example: Throwing from a Building
v₀ = 20 m/s, h₀ = 30 m
| Angle | Range (flat) | Range (from height) |
|---|---|---|
| 30° | 35.3 m | 60.2 m |
| 45° | 40.8 m | 62.8 m |
| 60° | 35.3 m | 55.4 m |
Note: The advantage of 45° diminishes with height.
Launching Below Target
When aiming upward at an elevated target:
- Need more initial velocity
- Different angle optimization
- May have no solution if target unreachable
Air Resistance Effects
Real projectiles don't follow perfect parabolas.
Why We Ignore Air Resistance
For basic calculations:
- Mathematics becomes very complex
- No closed-form solution
- Requires numerical methods
Good approximation when:
- Low speeds (< 20 m/s)
- Dense objects
- Short distances
Effects of Air Resistance
- Range decreases - drag slows horizontal motion
- Maximum height decreases - drag opposes upward motion
- Path is asymmetric - steeper descent than ascent
- Optimal angle decreases - often 35-40° instead of 45°
Drag Force
F_drag = ½·ρ·v²·C_d·A
Where:
- ρ = air density (1.225 kg/m³)
- v = velocity
- C_d = drag coefficient (0.3-0.5 for spheres)
- A = cross-sectional area
Terminal Velocity
When drag equals weight: v_terminal = √(2mg / (ρ·C_d·A))
Real-World Corrections
| Sport/Object | Range Reduction |
|---|---|
| Baseball | 10-15% |
| Golf ball | 40-50% |
| Shuttlecock | 80-90% |
| Bullet | Varies greatly |
Advanced Applications
Beyond basic trajectory calculations.
Ballistics
Interior ballistics: Acceleration in gun barrel Exterior ballistics: Flight through air (projectile motion + drag) Terminal ballistics: Impact effects
Corrections needed for:
- Coriolis effect (Earth's rotation)
- Altitude (air density changes)
- Temperature
- Wind
Sports Science
Optimal Strategies:
- Basketball free throw: ~52° angle
- Shot put: ~37° (arm mechanics limit angle)
- Soccer goal kick: ~45° for distance, flatter for speed
- Long jump: ~20° (limited by run-up)
Key insight: Biomechanical constraints often override theoretical optima.
Rocket Trajectories
For rockets, additional factors:
- Thrust vector (changes direction)
- Mass reduction (fuel consumption)
- Atmosphere exit (drag variation)
- Gravity variation with altitude
Video Game Physics
Games often use:
- Simplified drag models
- Adjustable gravity
- Non-parabolic "artistic" trajectories
- Unrealistic but fun physics
Pro Tips
- 💡Maximum range on flat ground occurs at 45° launch angle.
- 💡Two complementary angles (e.g., 30° and 60°) give the same range.
- 💡Horizontal and vertical motions are independent.
- 💡Time of flight depends only on vertical motion and gravity.
- 💡Horizontal velocity remains constant throughout flight (no air resistance).
- 💡Maximum height occurs when vertical velocity becomes zero.
- 💡For launches from height, optimal angle is less than 45°.
- 💡Air resistance always reduces range from the ideal calculation.
- 💡Mass doesn't affect trajectory (without air resistance).
- 💡Impact speed equals launch speed (same height, no air resistance).
- 💡The trajectory is always a parabola in ideal conditions.
- 💡Real sports trajectories are affected by spin (Magnus effect).
Frequently Asked Questions
The range formula R = v₀²·sin(2θ)/g is maximized when sin(2θ) = 1, which occurs at 2θ = 90°, or θ = 45°. At this angle, the balance between horizontal and vertical velocity components maximizes distance. However, this only applies to flat ground with no air resistance.
