Revision summary

State the force assumptions and coordinate directions first. Apply both components over the same time interval, and use the actual initial and final heights.

Weight and gravitational energy

Weight is gravitational force, not mass or automatically a support reading. With gravity alone, downward acceleration has magnitude g. The energy relation assumes a uniform field and uses vertical height change, positive for a rise and negative for a fall.

Derive the energy change from gravity's force times displacement and W_gravity = -ΔE_p. It refers to the mass-Earth system. A longer sloping path does not alter that gravitational change, although friction or a kinetic-energy change can alter the external work required.

Ideal projectile components

Assume uniform downward g, negligible drag and no continuing contact or launch force. With right/up positive:

The launch angle θ is measured above the positive horizontal. Horizontal acceleration is zero; vertical acceleration is -g throughout flight. At the top, only v_y is zero. A horizontal launch and a matched drop share a landing time because their vertical starting positions, velocities and accelerations match.

For video data, keep the camera and origin fixed, put the scale in the motion plane, track a consistent point and use actual frame times. Interval velocities belong at midpoint times for a uniform-acceleration model. Calibration and perspective errors are not removed by repetition.

Different landing heights

1. Put the actual landing height into the vertical position equation and select the future root.
2. Use that same time for horizontal range and both velocity components.
3. Combine the components for speed and use their signs for direction.
4. Check the speed with energy when gravity is the only force transferring mechanical energy.

For rightward motion, tan^-1(|v_y|/v_x) gives the angle's magnitude; the sign of v_y determines above or below horizontal. More generally, choose the quadrant from both component signs. Energy alone does not determine the flight time or direction.

The same-height time 2u_y/g is not the landing time when the landing level is different. A trajectory plot, a velocity-time plot and a force diagram describe different things.

Falling with drag

For a body released from rest in still air, with negligible buoyancy and unchanged drag conditions, upward drag grows as downward speed rises. With downward positive, F_resultant = mg - D and a = g - D/m.

At terminal speed, D = mg and a = 0. The body still moves, gravitational energy still decreases and energy still transfers to internal stores. Kinetic energy is constant. If drag conditions change so that D exceeds mg, the body can slow while still moving down.

Use definitions and units to distinguish the symbols: weight W is a force in N; W_gravity denotes work in J; W after a power value is the unit watt. N/kg and m/s² are equivalent units, but actual acceleration equals the gravitational field only when other forces are negligible.