Projectile motion
Throw it. Watch it fly. See every answer.
Projectile motion
Drag the launch arrow, or type the speed and angle. Slide the time to see the velocity at that moment. The dots are equal steps of time apart. Air resistance is ignored, as in NCERT.
A ball thrown at 20 m/s and 60 degrees rises 15 m, flies for 3.46 s and lands 34.64 m away.
Show the formulas
- Time of flight
- T = 2u sin θ / g = 2 × 20 × sin 60° / 10 = 3.46 s
- Range
- R = u² sin 2θ / g = 20² × sin 120° / 10 = 34.64 m
- Max height
- H = u² sin²θ / 2g = 20² × sin²60° / (2 × 10) = 15 m
- Longest range
- At θ = 45°, R max = u² / g = 40 m. θ and 90° - θ give the same range.
- Velocity
- vx = u cos θ = 10 m/s, vy = u sin θ - gt = 8.66 m/s at t = 0.87 s, so v = 13.23 m/s
- Path
- y = x tan θ - gx² / (2u² cos²θ), here y = 1.73x - 0.05x²
Pick a case
- Projectile motion formulas
- Range of a projectile
- Maximum height of a projectile
- Time of flight
- Equation of trajectory
- Horizontal projectile motion
- Horizontal projectile range formula
- Projectile from a height
- Projectile on an inclined plane
- Range on an inclined plane
- Maximum range at 45 degrees
- Complementary angles, same range
- Velocity at any time
- Projectile motion derivation
- Projectile at 30 degrees
- Projectile at 60 degrees
Same chapter: vectors, rectangular components.
Throws, paths and ranges
What are the formulas of projectile motion?
For a ball thrown at speed u and angle θ above the ground: time of flight T = 2u sin θ / g, range R = u² sin 2θ / g, maximum height H = u² sin²θ / 2g, and the path y = x tan θ - gx² / (2u² cos²θ).
Why is the path of a projectile a parabola?
The horizontal velocity u cos θ never changes, so x grows evenly with time. The vertical motion has a steady acceleration g downwards, so y has a t² term. Removing t leaves y in terms of x and x², which is a parabola.
At what angle is the range largest?
At 45 degrees, because R = u² sin 2θ / g and sin 2θ is largest, 1, when 2θ = 90 degrees. The longest range is then u² / g. Up an inclined plane of angle α it is 45 degrees + α/2.
Why do 30 and 60 degrees give the same range?
sin 2θ is the same for θ and 90 degrees - θ, since sin(180 degrees - 2θ) = sin 2θ. The steeper throw goes higher and stays up longer, but lands at the same spot.
What do the glowing dots on the path show?
Where the ball is after equal steps of time. Their feet on the ground are evenly spaced, because the horizontal speed is constant. Their heights bunch up near the top, because the vertical speed falls and then grows again.