Projectile Motion Calculator
Analyze parabolic motion, trajectory, range, and flight time
Calculate Projectile Motion
Speed at which projectile is launched
Angle above horizontal (45° = optimal range)
Height above ground at launch (0 = ground level)
Projectile Motion Results
Initial Velocity Components
Position at t = 1.4s
Physics Formulas Used
Velocity Components: vₓ = v₀cos(α), vᵧ = v₀sin(α)
Position: x = vₓt, y = h + vᵧt - ½gt²
Time of Flight: t = (vᵧ + √(vᵧ² + 2gh))/g
Range: R = vₓ × t
Max Height: h_max = h + vᵧ²/(2g)
Physics Analysis
Example: Basketball Free Throw
Problem Setup
Scenario: Basketball player shoots from free-throw line
Given: v₀ = 8 m/s, α = 45°, h = 2 m (release height)
Question: Will the ball reach the basket 4.6 m away?
Solution Steps
1. Calculate velocity components: vₓ = 8×cos(45°) = 5.66 m/s, vᵧ = 8×sin(45°) = 5.66 m/s
2. Find time of flight: t = (5.66 + √(5.66² + 2×9.81×2))/9.81 = 1.36 s
3. Calculate range: R = 5.66 × 1.36 = 7.7 m
4. Compare: 7.7 m > 4.6 m - Ball overshoots the basket!
Result: Need lower angle or less velocity for accurate shot
Real-World Examples
Basketball Shot
Free throw trajectory
Cannonball
Historical artillery
Soccer Ball Kick
Goal attempt
Arrow Shot
Archery target practice
Key Physics Concepts
Parabolic Trajectory
Curved path due to gravity
Velocity Components
Horizontal and vertical motion
Gravity Effect
Constant downward acceleration
Range Optimization
45° gives maximum range
Essential Formulas
Velocity Components
vₓ = v₀cos(α)
vᵧ = v₀sin(α)
Position Equations
x = vₓt
y = h + vᵧt - ½gt²
Range (Ground Level)
R = v₀²sin(2α)/g
Maximum at α = 45°
Maximum Height
h_max = h + v₀²sin²(α)/(2g)
Height above launch point
Understanding Projectile Motion
What is Projectile Motion?
Projectile motion describes the path of an object launched into the air, subject only to gravity. The trajectory forms a parabola due to the constant horizontal velocity and accelerating vertical motion.
Key Principles
Horizontal and vertical motions are independent. Horizontal velocity remains constant (no air resistance), while vertical motion follows free-fall physics with gravity acceleration of 9.81 m/s².
Optimal Launch Angle
For maximum range on level ground, the optimal launch angle is 45°. This balances horizontal distance with flight time. Different angles optimize for specific scenarios.
Real-World Applications
Sports (basketball, soccer, archery), military ballistics, space launches, and engineering design all use projectile motion principles for trajectory prediction and optimization.
Sports Applications
- • Basketball shooting angles
- • Soccer ball trajectories
- • Golf ball flight paths
- • Baseball pitching physics
Engineering Uses
- • Spacecraft launch trajectories
- • Artillery range calculations
- • Water fountain design
- • Safety equipment testing
Physics Education
- • Vector analysis
- • Kinematic equations
- • Energy conservation
- • Motion independence principle