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Past exam of the physics 1 paper of the University of Oxford / 2017 / 7 / ii / Solution by

Codex  0 2026-10-25

v_{x}

decays exponentially toward zero. The vertical velocity decreases monotonically toward

- m g / α

, with an initially steeper gradient

- g - (α / m) v_{0} sin θ

and positive second time derivative for an upward launch. It crosses zero at

t_{top} = \frac{m}{α} ln (1 + \frac{α v _{0} sin θ}{m g}) .

(1)

The right panel shows the two components and the formal terminal velocity. To identify the end of the physical flight, integrate the vertical velocity once:

y (t) = τ_{d} (v_{0} sin θ + g τ_{d}) (1 - e^{- t / τ_{d}}) - g τ_{d} t .

(2)

The first positive root of

y (t) = 0

is ground contact. The terminal velocity need not be reached before that event; dashed portions in the sketch are mathematical extensions without a ground surface.

Figure 1.
Projectile velocity components with and without linear drag
. An illustrative launch at $4 5^{\circ}$ , with dimensionless time $T = g t / v_{0}$ . The right panel uses $α v_{0} / (m g) = 1$ . Solid curves end at the first return to the launch height; dashed curves show the free-flight continuation. **Details**.

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