Twenty problems, three difficulty levels. Final answers at the end.
⭐ 1 (§4.1) — Particle at (-2, 3, 5). Position vector and magnitude?
⭐ 2 (§4.1) — From (1, 1) to (4, -3). Displacement vector, magnitude, angle?
⭐ 3 (§4.2) — \vec r(t) = 2t\hat i + t^2 \hat j. \vec v(t) and \vec a(t)?
⭐ 4 (§4.3) — Horizontal launch from 20 m at v_0 = 15 m/s. Landing time, range?
⭐⭐ 5 (§4.3) — Launch at v_0 = 30 m/s, \theta = 60°. Max height, range?
⭐⭐ 6 (§4.3) — Two objects released from the same window height simultaneously — one dropped, one thrown horizontally at 5 m/s. Which hits first?
⭐⭐ 7 (§4.3) — Throw at v_0 = 12 m/s to reach a ceiling 4 m above launch. Minimum angle?
⭐⭐ 8 (§4.4) — Car at 54 km/h on a curve of radius 50 m. a_c?
⭐⭐ 9 (§4.4) — Sphere of radius 0.3 m spinning at 3 rev/sec. Surface speed? a_c?
⭐⭐ 10 (§4.5) — Plane at 200 km/h north. Wind 50 km/h from the east. Ground velocity?
⭐⭐ 11 (§4.5) — Two cyclists: A east at 10 m/s, B north at 10 m/s. B's velocity relative to A?
⭐⭐ 12 (§4.3) — Fall from height H at rest. Using v = \sqrt{2gH}, impact speed from H = 45 m?
⭐⭐⭐ 13 (§4.3) — Launch at v_0 = 40 m/s, \theta = 30° from a 50 m cliff. Horizontal range on the flat below?
⭐⭐⭐ 14 (§4.3) — Show that the optimal launch angle from height h is \theta^* = \arctan(v_0/\sqrt{v_0^2 + 2gh}).
⭐⭐⭐ 15 (§4.4) — Satellite with period 95 min. Orbital radius in km? (GM = 4 \times 10^{14} m^3/s^2)
⭐⭐⭐ 16 (§4.5) — Boat at 5 m/s, river at 3 m/s. To land directly across at a 100 m-wide river, minimum time?
⭐⭐⭐ 17 (§4.3) — Soldier sees tank 1 km away. Rifle round at 500 m/s. What launch angle to hit (no air drag)?
⭐⭐⭐ 18 (§4.4) — Ball on a string of L = 1 m swings in a horizontal circle (conical pendulum). String angle 30° from vertical. Speed?
⭐⭐⭐ 19 (§4.2) — Particle with \vec v(t) = \omega(-\sin\omega t\hat i + \cos\omega t \hat j). Trajectory? Acceleration direction?
⭐⭐⭐ 20 (§4.5 + §4.3) — Boat on river flowing at 2 m/s. Ball thrown straight up inside the boat at 8 m/s. Where does it land?
Answers
1) \vec r = -2\hat i + 3\hat j + 5\hat k, magnitude \sqrt{38} \approx 6.16.
2) 3\hat i - 4\hat j, magnitude 5, angle -53.1°.
3) \vec v = 2\hat i + 2t\hat j, \vec a = 2\hat j.
4) t \approx 2.02 s, x \approx 30.3 m.
5) y_\text{max} \approx 34.4 m, R \approx 79.5 m.
6) Simultaneously — horizontal velocity doesn't affect vertical fall.
7) \sin\theta \geq \sqrt{2gh}/v_0 \Rightarrow \theta \geq 48.1°.
8) v = 15 m/s, a_c = 4.5 m/s^2.
9) v \approx 5.65 m/s, a_c \approx 106.6 m/s^2.
10) Total: -50\hat i + 200\hat j, magnitude \approx 206 km/h, ~14° west of north.
11) \vec v_{B/A} = -10\hat i + 10\hat j, magnitude 10\sqrt{2} \approx 14.1 m/s.
12) v = 29.7 m/s.
13) t \approx 5.83s, R \approx 202 m.
14) Prove by differentiating R(\theta) and setting to zero.
15) r \approx 6900 km.
16) For directly across: heading 36.9° upstream, speed 4 m/s, time 25 s.
17) \sin(2\theta) = 0.0392 \Rightarrow \theta \approx 1.12°.
18) v = \sqrt{gL\sin 30°\tan 30°} \approx 1.68 m/s.
19) Circular trajectory. \vec a points to the center — centripetal.
20) Ball lands back in the boat (like a projectile from a moving train — passenger sees it go straight up and down).
Preview of §4.8
Further resources for 2D/3D kinematics.
📚 See also: Halliday Vol 1, Ch 4 — Problems.
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