Twenty problems without step-by-step solutions — for working on your own. Original scenarios, difficulty varies:
- ⭐ Easy — single formula application
- ⭐⭐ Medium — multiple steps, choosing the right equation, careful sign handling
- ⭐⭐⭐ Hard — combining sections, heavier math, or subtle intuition
Each problem is tagged with the §2.x tools it exercises. Final answers at the end for self-check.
⭐ Problem 1 (§2.1)
A courier scooter runs back-and-forth between two buildings 800 m apart. In 5 minutes it completes the loop 3 times (out + back + out + back + out + back). Find:
a) total distance
b) total displacement (if it starts and ends at the origin)
c) average speed in m/s
⭐ Problem 2 (§2.2)
A particle's position: x(t) = 2t² - 5t + 3 (m, s).
a) Instantaneous velocity at t = 1 s and t = 3 s
b) When is the particle momentarily at rest?
c) Is acceleration constant or varying? What value?
⭐ Problem 3 (§2.3)
A car's velocity changes from 12 m/s at t = 5 s to -8 m/s at t = 15 s (negative means reversal).
a) Average acceleration
b) At approximately what time does the car momentarily stop? (Assume constant acceleration)
c) At t = 5, 10, 15 s, is the speed increasing or decreasing?
⭐⭐ Problem 4 (§2.4)
A cyclist at v_0 = 15 m/s starts accelerating at a = 0.5 m/s².
a) Velocity after 20 s?
b) How far in those 20 s?
c) If she aims for 25 m/s, how far from the start when she reaches that speed?
⭐⭐ Problem 5 (§2.4)
A car cruises at v_0 = 30 m/s. When the driver spots a hazard, she has 0.8 s reaction time (no braking, constant speed), then brakes at a = -6 m/s².
a) Distance during the reaction interval?
b) Braking distance?
c) Total stopping distance?
⭐⭐ Problem 6 (§2.5)
You throw a stone downward from a bridge with initial velocity v_0 = -3 m/s (negative because downward). The stone hits the water in 2.5 s.
a) Bridge height?
b) Impact speed?
c) If you had simply dropped it (v_0 = 0), when would it hit?
⭐⭐ Problem 7 (§2.5)
You throw a ball straight up from height y_0 = 2 m at v_0 = 12 m/s.
a) Maximum height
b) When does it return to the initial height (2 m)?
c) When does it hit the ground?
d) Ground-impact speed?
⭐⭐ Problem 8 (§2.6)
A cyclist's v-t diagram:
[0, 10]s:v = 4 + 0.5 tm/s (constant acceleration)[10, 30]s:v = 9m/s (constant)[30, 45]s:v = 9 - 0.6(t - 30)m/s (linear decrease)
a) Total distance?
b) Acceleration during [30, 45]s?
c) Sketch the x-t plot roughly (no need for precision, just recognize linear/parabolic segments).
⭐⭐ Problem 9 (§2.1 + §2.5)
You drop a rubber ball from 10 m. It bounces to 6 m, then to 3.6 m, and so on — each bounce reaches 0.6× the previous height. Assume each bounce is instantaneous.
a) Speed of the first impact?
b) Total distance until it stops? (Geometric series)
c) Total time until it stops? (Geometric time series)
⭐⭐ Problem 10 (§2.4 + §2.5)
You throw a stone upward from a cliff of unknown height H at v_0 = 20 m/s. It reaches the ground (bottom of the cliff) at t = 6 s.
a) Cliff height H?
b) Maximum height of the stone above the cliff top?
c) Impact speed?
⭐⭐ Problem 11 (§2.4)
Two cars start simultaneously from the same point, both from rest. Car A accelerates at a_A = 2 m/s². Car B at a_B = 3 m/s².
a) After 10 s, the distance between them?
b) At what time is the distance ratio x_B/x_A = 2?
⭐⭐ Problem 12 (§2.3)
A particle's position: x(t) = t³ - 6 t² + 9 t + 4 (m, s).
a) Differentiate for v(t) and a(t)
b) Find the turning points (rest instants)
c) At each turning point, is it reversing from + to − or the other way?
⭐⭐⭐ Problem 13 (§2.5 + §2.6)
You drop a ball from 10 m. Suppose air resistance gives it an extra acceleration −0.5 v (m/s² with v in m/s), always opposing the motion:
\[ a = -g - 0.5 v \]
(Signs with axis up positive.)
a) Terminal velocity (when a = 0)?
b) Using Euler's method with Δt = 0.1 s, compute velocity after 1 s approximately.
c) Without drag, what would v(1) be? What's the difference?
⭐⭐⭐ Problem 14 (§2.4 + §2.3)
A train starts from rest, accelerates at a = 1.2 m/s² until reaching 24 m/s, cruises at that speed for the rest of the 5-km segment, then brakes at -0.8 m/s² to a stop at station B.
a) Total travel time
b) Distance in each phase (accelerate, cruise, brake)
c) Sketch the v-t plot
⭐⭐⭐ Problem 15 (§2.2 + §2.6)
Velocity of a particle:
\[ v(t) = 3 t^2 - 12 t + 9\quad (\mathrm{m/s}) \]
with x(0) = 0.
a) Differentiate for a(t)
b) Find direction-reversal times (when v = 0)
c) Integrate to get x(t)
d) Compute the distance traveled in [0, 3]s (not displacement — watch for direction changes)
⭐⭐⭐ Problem 16 (§2.4 + §2.5)
You drop ball A from a 50 m building. After 0.5 s, you throw ball B straight down with initial speed v_0 = ? from the same point. Both hit the ground simultaneously.
Find v_0.
⭐⭐⭐ Problem 17 (§2.6)
A particle's acceleration:
\[ a(t) = 6 t - 2\quad (\mathrm{m/s^2}) \]
with initial conditions v(0) = 4, x(0) = 1.
a) Integrate to get v(t)
b) Integrate to get x(t)
c) At t = 5 s, what are position and velocity?
⭐⭐⭐ Problem 18 (§2.5)
You drop ball A from height h and at the same instant throw ball B up from ground level at v_0, directly below A. If we want them to meet at height h/2, what should v_0 be (in terms of h and g)?
⭐⭐⭐ Problem 19 (§2.4)
A driver at v = 25 m/s sees a hazard 50 m ahead. Reaction time is 0.8 s. What is the minimum braking |a| needed to stop just before the hazard?
⭐⭐⭐ Problem 20 (Kepler challenge — §2.3)
A satellite in a circular orbit around Earth at distance r = 7000 km from Earth's center. Orbital velocity v = 7.5 km/s. Compute the centripetal acceleration (formula a = v²/r) and compare with g_Earth at the surface. Is this consistent with the 1/r² law of gravity?
(Hint: g_surface = 9.8 m/s² at the surface, R_E = 6371 km. If the satellite is at r = 7000 km, the gravitational acceleration on it should be g_surface × (R_E/r)². Compare with v²/r.)
Final answers (for self-check)
1) a) 4800 m, b) 0, c) 16 m/s
2) a) v(1)=-1 m/s, v(3)=+7 m/s, b) t=1.25 s, c) constant a=4 m/s²
3) a) -2 m/s², b) t≈11 s, c) at t=5 slowing down, at t=10 speeding up (backward), at t=15 speeding up
4) a) 25 m/s, b) 400 m, c) 400 m
5) a) 24 m, b) 75 m, c) 99 m
6) a) 38.13 m, b) -27.5 m/s, c) 2.79 s
7) a) 9.35 m, b) 2.45 s, c) 2.60 s, d) -13.5 m/s
8) a) 273.75 m, b) -0.6 m/s², c) parabolic → linear → inverted parabola
9) a) 14 m/s, b) 40 m, c) 4.86 s (geometric series with ratio √0.6)
10) a) 56.4 m, b) 20.4 m (above cliff top), c) -38.8 m/s
11) a) 50 m, b) never (ratio is constant at 3/2)
12) a) v = 3t² - 12t + 9, a = 6t - 12, b) t=1, t=3, c) at t=1 from + to −, at t=3 from − to +
13) a) 19.6 m/s, b) v(1) ≈ -6.2 m/s (Euler), c) no drag: -9.8 m/s; difference ~5%
14) a) accel: 20 s (240 m), cruise: 208.33 s (5000 m), brake: 30 s (360 m). Total time: 258.33 s
15) a) a=6t-12, b) t=1, t=3, c) x=t³-6t²+9t, d) distance = 8 m (not Δx=0)
16) v₀ ≈ 9.35 m/s
17) a) v(t) = 3t² - 2t + 4, b) x(t) = t³ - t² + 4t + 1, c) v(5) = 69, x(5) = 121
18) v₀ = √(gh)
19) |a| ≈ 10.9 m/s² (assuming 30 m remain after reaction)
20) a_centripetal = 8.04 m/s²; gravity prediction = 9.8 × (6371/7000)² = 8.11 m/s². Yes, matches within ~1% — the 1/r² law is confirmed.
Preview of §2.9
Chapter 2 is closing. §2.9 gives further reading (parallel textbooks, videos, online experiments) in case anything remains unclear.
📚 See also: Halliday Vol 1, Ch 2 — Problems.
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