How Projectile Motion Works: Range, Height, and Angle

Quick Answer

Projectile motion is the curved path of an object thrown or launched under gravity alone. Three inputs set that path: the launch speed, the launch angle, and gravity. The range is R = v squared times sin(2 theta) divided by g. For example, a launch at 25 m/s and 45 degrees on Earth lands about 64 metres away. What angle gives the maximum range is always 45 degrees, because that splits the launch speed evenly between forward reach and time in the air. Open the simulator, drag from the launcher to aim, and watch the range, peak height, and time of flight update live.

How Projectile Motion Works

The key idea in this corner of kinematics is that horizontal and vertical motion are independent. Gravity only pulls downward, so it changes the vertical velocity while the horizontal velocity stays constant during the whole flight. The projectile therefore drifts sideways at a steady rate while it rises, slows, stops at the peak, and falls. Combine the two and the path is a symmetric parabola. For example, watch the velocity arrow in the simulator: its horizontal length never changes, but its vertical part shrinks to zero at the top and grows again on the way down.

Why 45 Degrees Gives the Maximum Range

The projectile range depends on sin(2 theta), and the sine function is largest at 90 degrees, so the range peaks when twice the angle is 90, meaning the angle is 45 degrees. Steeper than that and the projectile spends too much of its speed climbing; shallower and it runs out of airtime before it covers ground. For example, set the speed to 25 m/s and sweep the angle slider: the landing marker reaches furthest exactly at 45 degrees and falls off on either side. To calculate the range of a projectile by hand, square the speed, multiply by the sine of twice the angle, then divide by gravity.

How Launch Angle Trades Distance for Height

Complementary launch angles compared this way, two angles that add up to 90 degrees such as 30 and 60, give the same range but very different flights. The steeper launch reaches a higher peak and stays airborne longer, while the shallower one is faster and lower. For example, set 60 degrees and note the tall, slow arc, then set 30 degrees and watch a flat, quick shot land in the same spot. This is why a lofted pass and a driven pass can cover the same distance while looking nothing alike.

What Gravity Changes

Gravity sets how quickly the projectile is pulled back down, so it controls both the peak height and the flight time. Weaker gravity lets the projectile hang longer and travel much further from the same launch, because range scales with 1 divided by g. For example, load the Moon preset, where gravity is about one sixth of Earth, and the same speed and angle send the projectile several times as far. Gravity never touches the horizontal velocity, only the vertical part of the motion.

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