Parametric Projectile Motion Simulator — Theory
1. Projectile motion as parametric equations
Projectile motion is naturally described by parametric equations because the horizontal and vertical
coordinates both depend on time.
\[
x=x(t),
\qquad
y=y(t).
\]
The parameter is time \(t\).
2. Velocity components
If the initial speed is \(v_0\) and the launch angle is \(\theta\), then
\[
v_{0x}=v_0\cos\theta,
\]
\[
v_{0y}=v_0\sin\theta.
\]
These components split the motion into horizontal and vertical parts.
3. Ideal projectile motion
Without air resistance, horizontal velocity is constant and vertical acceleration is \(-g\). The equations are
\[
x(t)=x_0+v_0\cos\theta\,t,
\]
\[
y(t)=y_0+v_0\sin\theta\,t-\frac12gt^2.
\]
These are the basic parametric equations of projectile motion.
4. Time of flight
The time of flight is found by setting the vertical position equal to the ground level:
\[
y_0+v_0\sin\theta\,t-\frac12gt^2
=
y_{\text{ground}}.
\]
This gives a quadratic equation in \(t\). The physical flight time is the positive solution.
5. Range
The horizontal range is the horizontal displacement at landing:
\[
R=x(T)-x_0.
\]
For ideal motion with equal launch and landing height,
\[
R=\frac{v_0^2\sin(2\theta)}{g}.
\]
This formula is exact only for ideal motion and equal launch and landing height.
6. Maximum height
The projectile reaches maximum height when the vertical velocity becomes zero:
\[
v_y(t)=v_{0y}-gt.
\]
In ideal motion, the time to peak is
\[
t_{\max H}=\frac{v_{0y}}{g},
\]
when \(v_{0y}>0\). The maximum height is
\[
H_{\max}=y_0+\frac{v_{0y}^2}{2g}.
\]
7. Linear air resistance
Linear drag assumes the drag force is proportional to velocity:
\[
\vec F_d=-b\vec v.
\]
Dividing by mass gives acceleration:
\[
\vec a=\langle0,-g\rangle-\frac{b}{m}\vec v.
\]
This model is useful when drag is approximately proportional to speed.
8. Quadratic air resistance
Quadratic drag assumes the drag force is proportional to speed squared and points opposite the velocity:
\[
\vec F_d=-c\lVert\vec v\rVert\vec v.
\]
The acceleration becomes
\[
\vec a=\langle0,-g\rangle-\frac{c}{m}\lVert\vec v\rVert\vec v.
\]
This model is common for faster projectiles moving through air.
9. Numerical simulation
With air resistance, the motion usually cannot be described by the simple ideal equations. Instead, the
calculator solves the system
\[
\frac{dx}{dt}=v_x,
\qquad
\frac{dy}{dt}=v_y,
\]
\[
\frac{dv_x}{dt}=a_x,
\qquad
\frac{dv_y}{dt}=a_y.
\]
The calculator uses fourth-order Runge-Kutta integration for the drag models.
10. Optimal launch angle
In ideal motion from ground to ground, the maximum range occurs at
\[
\theta=45^\circ.
\]
But this is not always true. The best angle changes when:
- the launch height is not equal to the landing height;
- air resistance is present;
- gravity or drag coefficients change;
- the projectile starts above or below the ground level.
The calculator estimates the best angle by comparing many launch angles numerically.
11. Units on graph axes
Projectile motion represents physical distance. Therefore, graph axes should show units:
\[
x\;(\mathrm{m}),
\qquad
y\;(\mathrm{m}).
\]
The distance unit can be changed to feet, centimeters, or another unit. The same unit should be used
consistently for height, range, and gravity.
13. Common mistakes
- Mixing units: if \(g=9.8\), distances should usually be meters and time seconds.
- Using degrees inside calculus formulas: trigonometric formulas use radians mathematically, but the calculator can convert degrees.
- Assuming the best angle is always \(45^\circ\): this is only true in the ideal level-ground case.
- Ignoring launch height: starting above the ground changes the time of flight and range.
- Using too large a time step: drag simulations are numerical, so smaller \(\Delta t\) gives smoother results.
- Forgetting that drag opposes velocity: drag is not always straight downward; it points opposite the direction of motion.
