Launch It Yourself: How Projectiles Fly

How to use this simulation

1. Move the velocity and angle sliders, or drag on the stage to aim directly: the drag direction sets the angle, its length sets the speed.
2. Keep "earlier launches" on and fire at 30° and then 60°: the two land at the same range.
3. Switch worlds: with the Moon's weaker g the same throw goes much farther and higher.
4. The green arrow (vₓ) never changes; the red one (vᵧ) shrinks to zero at the top and flips.
5. Slow it to 0.25× and pause at any moment to copy the readings into your notebook.

Every throw you have ever made was a projectile

Watch a long pass in football, a basketball free throw or a golf drive. None of them travels in a straight line. The ball rises, seems to hang for a moment, then curves back down. That graceful curve is what this page is about.

Think about tossing your keys to a friend across the room. Throw straight up and they fall back into your own hand. Throw flat and they hit the floor halfway. Without doing any maths you pick an angle, a bit up and a bit forward. That instinct is projectile physics.

A garden hose, a fountain, a frog leaping, a paper ball aimed at the bin: each follows the same rules. Once you know those rules you can predict, before the throw, how high the object will rise, where it will land and how long it will stay in the air.

Starting from zero: two motions that ignore each other

The single best trick in this chapter is to split the motion into two parts: horizontal (sideways) and vertical (up and down). The surprising part is that they do not affect each other at all. Each does its own thing, and together they draw the curve.

Horizontally, once the ball leaves your hand nothing pushes it forward and nothing pulls it back (ignore air resistance for now). Newton's first law says that with no force the velocity stays the same. So the horizontal velocity is constant, and the ball covers equal distances sideways every second.

Vertically, the Earth pulls down the whole time. The upward speed drops by about 9.8 m/s every second until it hits zero at the very top, then the ball starts falling and its downward speed grows by 9.8 m/s every second.

Put them together: steady forward progress plus a rise-and-fall. Distance forward grows in step with time, height changes with the square of time, and a curve built that way is called a parabola. That is why every projectile path (without air resistance) is a parabola.

Try this thought experiment. From the same height, drop one ball and at the same instant flick another sideways. Which lands first? Both land together. Falling is a purely vertical matter, and vertically both start at zero speed with the same g. The sideways speed only carries the second ball farther; it does not change how fast it falls.

Key terms at a glance

Definitions first; most "what is…" exam questions come straight from this table.

Deriving the formulas step by step

If you can derive the formulas yourself you will never mix them up in an exam. Take a body launched from the ground at speed v₀ and angle θ, with no air resistance, and take upward as positive.

Experiments to try in the simulation

The animation above is your pocket laboratory. Run these one at a time, and before each one guess the result, then check.

• Experiment 1: keep the speed at 20 m/s and launch at 15°, 30°, 45°, 60° and 75°. Which angle goes farthest?
• Experiment 2: with "keep earlier launches" on, compare 30° and 60° side by side. The two paths land on the same point.
• Experiment 3: keep the angle and double the speed from 10 to 20 m/s. Does the range double or quadruple? The formula has v₀², so it should quadruple.
• Experiment 4: move from Earth to the Moon. How many times farther does the same throw go? What happens on Jupiter?
• Experiment 5: raise the launch height h₀. Now 45° is no longer the best angle; from a height a slightly lower angle goes farther.
• Experiment 6: pause at the top and look for the red arrow (vᵧ). It has vanished, but the green arrow (vₓ) is still there.

Solved problems

Now with numbers. Each solution is worked step by step, and you can check every one by entering the values in the simulation. g = 9.8 m/s² throughout.

What changes with the angle: a quick table

Values for v₀ = 20 m/s and g = 9.8 m/s² at five angles. Compare the ranges of the pairs 15° and 75°, and 30° and 60°.

• A bigger angle always means a greater height and a longer flight.
• The range grows up to 45° and then shrinks, and complementary angles share a range.

Mistakes almost everyone makes

A few errors come up again and again in this chapter. Knowing them saves easy marks in multiple-choice questions.

• Wrong: "the velocity is zero at the top." Right: only the vertical velocity is zero; the horizontal velocity v₀cosθ is still there.
• Wrong: "the acceleration is zero at the top." Right: the acceleration is g downward everywhere on the path, including the top.
• Wrong: "a heavier ball goes less far." Right: without air resistance, mass appears in none of the formulas.
• Wrong: treating sin2θ as 2sinθ. sin2θ = 2sinθcosθ; sin90° = 1 but 2sin45° ≈ 1.414.
• Wrong: using T = 2v₀sinθ/g for a launch from a height. That formula only holds when the body lands at its launch height.

Projectiles in real life

Sport is full of projectiles. Corner kicks, basketball shots, the javelin and the shot put: athletes train until they feel the right angle. A shot put leaves from shoulder height, so its best angle is a little below 45°, the effect you saw by raising h₀ in the simulation.

Firefighters choose the angle of a hose by the reach of the water, and fountain designers match the water speed and nozzle angle to shape their arches.

Real air pushes back. Drag slows the body in both directions, so the real path is not a perfect parabola: the descent is steeper, the range shorter and the best angle slips below 45°. Light, bulky objects such as a badminton shuttlecock feel this most. Textbook formulas describe the ideal, drag-free case.

Exam corner

Projectile questions follow a few familiar patterns. Here is what to expect and how to answer.

Quick revision summary

Read just this before an exam and the whole chapter comes back.

• Projectile: thrown at an angle, moving under gravity alone; the path is a parabola.
• vₓ = v₀cosθ is constant; vᵧ = v₀sinθ − gt changes; acceleration is always g downward.
• T = 2v₀sinθ/g, H = v₀²sin²θ/2g, R = v₀²sin2θ/g.
• The range is greatest at 45° (v₀²/g); complementary angles share a range.
• Thrown horizontally from height h: t = √(2h/g), x = u·t.
• Weaker g (the Moon) raises T, H and R; mass appears in no formula.

Frequently asked questions