Projectile Motion
Explore the physics of objects launched at an angle — trace trajectories, analyze velocity components, and understand range and height in real time.
Key Notes
Projectile motion is a combination of two independent motions: uniform horizontal motion and uniformly accelerated vertical motion (under gravity).
The horizontal component of velocity (uₓ = u cos θ) remains constant throughout the flight (no air resistance).
The vertical component of velocity (uᵧ = u sin θ) changes due to gravitational acceleration g = 9.8 m/s².
At the highest point, the vertical velocity is zero, but horizontal velocity persists — the object is NOT at rest.
The trajectory of a projectile is a parabola described by: y = x tan θ − (gx²)/(2u²cos²θ).
Time of flight, maximum height, and range depend only on initial speed and launch angle (in vacuum).
The motion is symmetric about the highest point — time of ascent equals time of descent.
Projectile motion analysis is a cornerstone of JEE Mechanics and connects to relative motion, energy, and circular motion concepts.
Formulas
Time of Flight
Total time the projectile stays in the air.
Maximum Height
The highest point reached by the projectile.
Range
Horizontal distance covered. Maximum at θ = 45°.
Trajectory Equation
Parabolic path equation.
Velocity at time t
Magnitude of velocity at time t.
Angle at time t
Direction of velocity vector at any instant.
Important Points
At θ = 45°, the range is maximum for a given speed. Complementary angles (e.g., 30° and 60°) give the same range.
The velocity at the highest point equals u cos θ (horizontal component only) — it is NOT zero.
Air resistance reduces range and makes the trajectory asymmetric — JEE typically assumes vacuum.
For projectiles from a height, use quadratic formula for time of flight (vertical displacement ≠ 0).
Radius of curvature at highest point: Rc = u²cos²θ / g — frequent JEE question.
Energy conservation applies: KE + PE = constant. At the top, KE is minimum (not zero) and PE is maximum.
On inclined planes, resolve g along and perpendicular to the incline for cleaner analysis.
Projectile Motion notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.