Projectile Motion

Projectile motion is one of the first problems you encounter in classical mechanics — and one of the most elegant. An object is launched into the air with some initial speed $v_0$ at an angle $\theta$ above the horizontal. After that moment, the only force acting on it is gravity (we ignore air resistance).

Because gravity acts only downward, the motion separates neatly into two independent components:

Component	Acceleration	Velocity	Position
Horizontal (x)	$0$	$v_{0x} = v_0\cos\theta$ (constant)	$x = v_0 \cos\theta\; t$
Vertical (y)	$-g$	$v_{0y} = v_0\sin\theta - g\,t$	$y = v_0 \sin\theta\; t - \tfrac{1}{2}g\,t^2$

Combining these gives the parabolic trajectory:

$x(t) = v_0 \cos\theta \; t \qquad y(t) = v_0 \sin\theta \; t - \tfrac{1}{2}g\,t^2$

Key Derived Quantities

From these equations we can derive three important results (assuming level ground, $y_0 = 0$):

• Time of flight: $\displaystyle T = \frac{2\,v_0 \sin\theta}{g}$
• Maximum height: $\displaystyle H = \frac{v_0^2 \sin^2\theta}{2g}$
• Range: $\displaystyle R = \frac{v_0^2 \sin 2\theta}{g}$

Notice that the range depends on $\sin 2\theta$, which is maximised when $2\theta = 90°$, i.e. $\theta = 45°$. So for any given launch speed, a 45° angle gives the longest range (in the absence of air resistance).

Symmetry insight: Launch angles that add up to 90° (e.g. 30° and 60°) give the same range but very different trajectories. Try it below!

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What to Observe

• Angle vs Range: Set velocity to 50 m/s. Sweep the angle from 10° to 80°. Notice the range peaks at 45°, and complementary angles (e.g. 30° & 60°) give the same range but different peak heights.
• Gravity's role: Lower gravity (think Moon at ~1.62 m/s²) dramatically increases both range and height. On Jupiter (~24.8 m/s²), the same throw barely gets off the ground.
• Speed matters quadratically: Doubling $v_0$ quadruples the range ($R \propto v_0^2$). That's why a small increase in launch speed makes a huge difference.

Real-World Applications

• Sports: Every ball sport involves projectile motion — from football goal-kicks to basketball free throws and cricket sixes.
• Artillery & Rocketry: Ballistic trajectories were the original motivation for studying this problem (Galileo, 1638).
• Space launches: Orbital mechanics begins where projectile motion meets the curvature of the Earth.

Limitations of This Model

This simulation assumes no air resistance. In reality, drag force ($F_d = \tfrac{1}{2} C_d \rho A v^2$) slows the projectile, reduces the range, and makes the trajectory asymmetric — the descending arc is steeper than the ascending one.