⚡ General Physics & Classical Mechanics

Projectile Motion Calculator

Q: What launch angle produces the maximum range in projectile motion?

When launched from ground level (initial height y0 = 0) with no air resistance, a launch angle of 45 degrees yields the maximum horizontal range because sin(2θ) reaches its maximum value of 1 at 2(45°) = 90°.

Q: Why is horizontal velocity constant in ideal projectile motion?

Assuming negligible air resistance, gravity acts solely in the vertical downward direction. Because there is no horizontal acceleration (ax = 0), the horizontal velocity vx remains constant throughout flight: vx = v0 cos(θ).

Simulate 2D projectile kinematics with real-time trajectory plotting. Calculate total range, flight time, maximum apex height, and instantaneous velocity vectors.

Presets:

Launch Parameters

Initial Velocity (v₀) m/s

Launch Angle (θ) 45°

Initial Height (y₀) meters

Gravitational Acceleration (g) m/s²

Total Range (R)

63.71 m

Horizontal distance

Max Height (H)

15.93 m

Peak above ground

Flight Time (t)

3.60 s

Total hang time

Impact Speed

25.00 m/s

Final magnitude

🎯 Real-Time Flight Trajectory

y(x) parabolic arc

📝 Kinematics Step-by-Step Derivations

Physics Behind Projectile Motion

Projectile motion is the two-dimensional motion of an object thrown or projected into the air, subject only to acceleration as a result of gravity. The horizontal and vertical motions are completely independent of each other:

Horizontal motion (x-axis): Zero acceleration (a_x = 0). Constant velocity v_x = v₀ cos(θ).
Vertical motion (y-axis): Constant downward acceleration (a_y = -g). Governed by SUVAT kinematic equations.

Core Governing Equations

Maximum Apex Height (H)

H = y₀ + (v₀ sin θ)² / (2g)

Calculated where vertical velocity component vy momentarily equals zero.

Horizontal Range (R)

R = v₀ cos(θ) × t_flight

For flat ground (y₀ = 0), simplifies to R = (v₀² sin 2θ) / g.

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Projectile Motion Calculator

Launch Parameters

🎯 Real-Time Flight Trajectory

📝 Kinematics Step-by-Step Derivations

Physics Behind Projectile Motion

Core Governing Equations

Maximum Apex Height (H)

Horizontal Range (R)

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