Projectile motion formulas
For a projectile launched from the ground with speed u at angle θ, ignoring air resistance:
| Quantity | Formula |
|---|---|
| Time of flight | T = 2u sin θ / g |
| Maximum height | H = u² sin²θ / 2g |
| Horizontal range | R = u² sin 2θ / g |
| Equation of path | y = x tan θ − g x² / (2u² cos²θ) |
Facts JEE and NEET like to test
- Range is greatest at 45°, and Rmax = u²/g.
- Complementary angles give the same range. 30° and 60° land at the same spot, but 60° goes higher and stays up longer.
- At the top, the velocity is u cos θ (horizontal only), not zero. The acceleration is still g, downwards.
- tan θ = 4H/R. If range equals 4 × max height, the angle is 45°.
For a launch from a height, the time of flight comes from the quadratic h + u sin θ · t − ½gt² = 0. The calculator handles that for you. All the kinematics formulas are on the physics formula sheet.
Questions students ask
At what angle is the range maximum?
45°, when launched and landing at the same height. Then R = u²/g.
What is the velocity at the highest point of a projectile?
u cos θ, in the horizontal direction. The vertical part is zero at the top.
Which angles give the same range?
Any two angles that add up to 90°, like 30° and 60° or 15° and 75°.
Should I use g = 10 or 9.8?
Use what the question says. JEE and NEET papers usually say g = 10 m/s². If nothing is given, 9.8 is safer.