Projectile Motion Simulator

What is projectile motion?

Throw a ball and it does not fly in a straight line and it does not drop straight down. It sweeps out a smooth curve, up and over and back to the ground. That curve is what projectile motion is about, and there is one neat idea that explains all of it.

The idea is that the ball is really doing two things at once, and they do not interfere. Sideways, it just keeps moving at a steady speed, because nothing is pushing or slowing it in that direction. Up and down, gravity is at work: it slows the ball as it rises, stops it at the top, then speeds it up again as it falls.

Stack those two motions together and you get the arc. Steady sideways drift plus rise-and-fall equals a curve called a parabola. Every unpowered thing that flies makes one, whether it is a basketball, a water fountain or a cannonball.

The angle you launch at decides the shape. Throw almost flat and it does not go far. Throw almost straight up and it comes back near where it started. On flat ground the sweet spot for distance is right in the middle, 45 degrees. The simulator at the top lets you set the speed, angle and height and watch the arc and the numbers change.

How do you calculate projectile motion?

Split the launch velocity into two pieces. If the speed is \(v_0\) at an angle \(\theta\) above the horizontal, the horizontal part is \(v_x = v_0\cos\theta\) and the vertical part is \(v_y = v_0\sin\theta\). The key fact is that these evolve separately.

Horizontally there is no force (ignoring air), so \(v_x\) is constant and \(x = v_0\cos\theta\,\cdot t\). Vertically, gravity gives a constant downward acceleration \(g\), so the height follows the familiar equation of motion

\[ y = h + v_0\sin\theta\,\cdot t - \tfrac12 g t^2, \]

where \(h\) is the launch height. The projectile reaches its peak when the vertical velocity hits zero, at time \(t = v_0\sin\theta / g\), giving a maximum height of \(h + (v_0\sin\theta)^2 / (2g)\).

Setting \(y = 0\) and solving the quadratic gives the flight time, and multiplying by \(v_x\) gives the range. For a launch and landing at the same height, the range simplifies to

\[ R = \frac{v_0^2 \sin 2\theta}{g}. \]

Because \(\sin 2\theta\) is largest at \(2\theta = 90^\circ\), the range is greatest at \(\theta = 45^\circ\), and complementary angles like \(30^\circ\) and \(60^\circ\) give the same distance. Launch from a height and that neat symmetry shifts: the best angle drops below 45 degrees, which the simulator shows if you raise the launch point.

The kinematics of a projectile trajectory

Independence of components

Projectile motion is the cleanest example of vector superposition in kinematics. With acceleration \(\mathbf{a} = (0, -g)\), the equations for \(x\) and \(y\) decouple completely: the horizontal coordinate is uniform motion and the vertical is uniform acceleration, and eliminating \(t\) between them gives \(y = h + x\tan\theta - \frac{g x^2}{2 v_0^2 \cos^2\theta}\), the equation of a parabola. Galileo's insight that the two axes are independent, demonstrated by dropping and projecting bodies together, was the conceptual break that made a quantitative science of motion possible.

Range, height and the launch angle

For launch and landing at equal height the range \(R = v_0^2 \sin 2\theta / g\) peaks at \(45^\circ\). From a nonzero height \(h\) the optimum angle is \(\theta^\* = \arctan\!\big(v_0 / \sqrt{v_0^2 + 2gh}\big)\), always less than \(45^\circ\) and falling as \(h\) grows. Maximum height and range trade off through the same speed budget: all the energy going into \(v_y\) buys altitude, all into \(v_x\) buys distance, and \(45^\circ\) splits them for the best reach on level ground.

Why real trajectories are not parabolas

The parabola assumes no air. Real drag is roughly proportional to \(v^2\) and opposes the velocity, which shortens the range, lowers the optimal angle (often to the mid-30s for a thrown ball), and makes the descending side of the path steeper than the ascending side, a skewed curve rather than a symmetric arc. Spin adds a sideways Magnus force, and over long ranges the Earth's curvature and rotation (the Coriolis effect) matter, which is why artillery and ballistics use numerical models rather than the closed-form parabola.

The same maths as orbits

Projectile motion is the local, flat-ground limit of orbital motion. Newton's cannonball makes the link: fire fast enough and the ground curves away as quickly as the ball falls, and the parabola closes into an ellipse, an orbit. Uniform gravity giving a parabola is the small-scale approximation to the inverse-square gravity that gives Kepler's ellipses. A thrown ball and a moon are doing the same thing at different scales.