What is projectile motion?
A projectile is any object that is launched and then moves only under the influence of gravity (and, in real life, air resistance). Once it leaves the cannon, nothing pushes it forward any more. Its path, called thetrajectory, is a parabola when air resistance can be ignored.
Two independent motions
The key idea, first described clearly by Galileo, is that horizontal and vertical motion are independent of each other:
- Horizontally, there is no force (without air), so the projectile moves at a constant speed vₓ = v cos θ.
- Vertically, gravity gives a constant downward acceleration g. The vertical velocity starts at v sin θ, decreases to zero at the top, and then grows in the downward direction.
Because the two motions are independent, a ball fired horizontally from a cliff hits the ground at the same moment as a ball simply dropped from the same height — the horizontal speed does not change how quickly it falls.
Formulas
For launch speed v, angle θ above horizontal, launch height h and gravitational acceleration g:
x(t) = v·cos θ · t y(t) = h + v·sin θ · t − ½·g·t²On level ground (h = 0):
Flight time T = 2v·sin θ / g Range R = v²·sin 2θ / g Max height H = v²·sin²θ / 2gWith a launch height, the flight time is T = (v sin θ + √((v sin θ)² + 2gh)) / g. The simulator uses this to place the predicted landing triangle.
Launch angle and range
On level ground without air resistance, 45° gives the greatest range, and complementary angles such as 30° and 60° give exactly the same range (one flat and fast, one high and slow). Launching from a height changes this: the best angle becomes slightly less than 45°, because the ball already has extra time to travel as it falls.
Air resistance
Real projectiles push air out of the way, which creates a drag force opposite to their motion. For fast-moving balls, drag grows roughly with the square of speed. Drag slows the projectile both horizontally and vertically, so the range and maximum height shrink, the descent becomes steeper than the climb, and the best launch angle drops below 45°. Switch on air resistance and compare with the dashed no-air prediction.
Key terms
- Trajectory
- The path a projectile follows through space.
- Velocity components
- The horizontal (vₓ) and vertical (v_y) parts of the velocity, found with cosine and sine of the launch angle.
- Range
- The horizontal distance travelled before landing.
- Apex
- The highest point of the trajectory, where vertical velocity is zero.
- Acceleration due to gravity (g)
- About 9.81 m/s² on Earth, 1.62 m/s² on the Moon, 3.71 m/s² on Mars and 24.8 m/s² on Jupiter’s cloud tops.
- Drag
- The resistive force from air or another fluid, acting opposite to the direction of motion.
Real-world applications
- Sports: basketball shots, long jump and shot put all involve choosing a good launch angle and speed. Athletes usually launch below 45° because they release from a height and must fight air resistance.
- Engineering and safety: fountain designers, firefighters aiming hoses and engineers planning debris zones use projectile equations.
- Space exploration: engineers account for lower gravity when planning how rovers and astronauts move rocks or equipment on the Moon and Mars.
- Forensics: investigators reconstruct trajectories to work out where an object was thrown from.
Common misconceptions
- “A force keeps pushing the ball forward.” After launch, only gravity (and drag) act. The forward motion continues because of inertia, not a force.
- “At the top, the ball stops.” Only the vertical velocity is zero; the ball still moves horizontally.
- “Heavier objects fall faster.” Without air resistance, all objects have the same acceleration g regardless of mass.
- “45° is always best.” Only on level ground with no air resistance.
What the model simplifies
The ball is treated as a point with no spin, so effects such as the curve of a spinning football (the Magnus effect) are left out. Gravity is treated as constant and the ground as flat. Air resistance uses one fixed drag coefficient, roughly that of a tennis ball, and assumes still air with no wind.
Key takeaways
- Projectile motion is two independent motions: constant velocity horizontally and constant acceleration vertically.
- Without air resistance, the trajectory is a parabola and 45° gives the maximum range on level ground.
- Range is proportional to v² and inversely proportional to g.
- Air resistance shortens the path, makes it asymmetric and lowers the best launch angle.