Kinematics

Final velocity in terms of initial velocity, uniform acceleration, and time.

Position of a body moving at constant velocity.

Displacement per unit time over a finite interval.

Displacement in terms of initial velocity, acceleration, and time.

Change in velocity per unit time over a finite interval.

Relates final velocity to initial velocity, acceleration, and displacement.

Total time a projectile remains in the air when launched from and landing on level ground.

Velocity at a specific instant — the derivative of position with respect to time.

Acceleration at a specific instant — the derivative of velocity with respect to time.

Greatest vertical height reached by a projectile launched from level ground.

Horizontal range on level ground. Maximum range at θ = 45°.

Velocity of A as observed from B.

Horizontal and vertical velocity components at any time t.

Magnitude of velocity vector at any instant during projectile motion.

Direction of velocity vector with respect to horizontal at any instant.

Horizontal and vertical coordinates of a projectile at time t.

Path of a projectile — a downward-opening parabola.

Time for a horizontally projected body to fall height h.

Horizontal distance covered by a body projected horizontally from height h.

Speed of a horizontally projected body at time t.

Parabolic path of a horizontally projected body.

Time of flight when projected at angle θ from height h above ground.

Horizontal range when projected at angle θ from height h.

Equation of path when projected at angle θ from height h.

Range along an inclined plane when projected up the slope. α = projection angle from horizontal, β = inclination angle.

Range along an inclined plane when projected down the slope.

Maximum range achievable on an inclined plane of angle β.

Velocity of a projectile launched from a moving body — vector addition of body velocity and launch velocity.

Air resistance reduces range, maximum height, and time of flight. Optimal angle shifts below 45°.