Complete Theory

Worked Examples

Example 1Train — passenger throws a ball vertically

Example 2Boat in a river

Example 3Coriolis force on Earth

Exercises with Solutions

Exercise 1Velocity composition — airplane with crosswindMedium

📋 Problem to solve

A pilot wants to fly due north. The airplane's airspeed is

v_{plane/air} = 300\,\mathrm{m/s}

. A crosswind blows from west to east at

v_{wind} = 40\,\mathrm{m/s}

. (a) If the pilot simply points north, what is the actual ground speed and direction? (b) What heading angle must the pilot maintain to fly exactly north?

📌 Given data

v_{plane/air} = 300 m/s north (relative velocity to air S')v_{wind} = 40 m/s east (velocity of air S' relative to ground S)

Exercise 2Accelerating frame — apparent weight in an elevatorHard

📋 Problem to solve

A person of mass

m = 70\,\mathrm{kg}

stands on a scale inside an elevator. The elevator accelerates upward at

a = 3\,\mathrm{m/s^2}

. (a) What does the scale read (apparent weight)? (b) What fictitious force acts on the person in the elevator's frame? (c) If the elevator accelerates downward at

3\,\mathrm{m/s^2}

, what does the scale read? (d) What happens in free fall (

a = g

📌 Given data

m = 70\,\mathrm{kg}

(person's mass)

a_{elev} = 3\,\mathrm{m/s^2}

upward (acceleration of S' = elevator relative to S = ground)

📚

Recommended Books

Introductory

Physics for Scientists and Engineers

Goldstein, Poole, Safko

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Integrative Problems

Problems combining all chapters — exam level

Problem 1Tower, Ballistic Pendulum, and Keplerian OrbitEXTREME

A cannon is placed on top of a tower

h_0 = 50\,\mathrm{m}

tall and fires a projectile of

m = 0.025\,\mathrm{kg}

horizontally at

v_0 = 400\,\mathrm{m/s}

.

The projectile strikes and embeds in a wooden block

M = 4.0\,\mathrm{kg}

hanging from a rope of length

L = 2.0\,\mathrm{m}

(ballistic pendulum), at ground level.

The Earth-Moon system is then used as a reference for Kepler's third law.

📌 Problem data

h_0 = 50\,\mathrm{m}

m = 0.025\,\mathrm{kg}

v_0 = 400\,\mathrm{m/s}

M = 4.0\,\mathrm{kg}

L = 2.0\,\mathrm{m}

(a)Uniformly Accelerated Motion(b)Inelastic Collision(c)Potential Energy + Pendulum(d)Moment of Inertia — Rigid Body(e)Gravitation — Kepler's Third Law

Problem 2Spring, Rolling Disk, Inclined Plane Collision, and ConservationEXTREME

A spring (

k = 6000\,\mathrm{N/m}

, compressed

x_0 = 0.25\,\mathrm{m}

) launches a solid disk (

M = 3.0\,\mathrm{kg}

R = 0.15\,\mathrm{m}

) up an inclined plane (

\theta=30°

L=5\,\mathrm{m}

\mu_d=0.06

) that rolls without slipping.

At the top the disk is launched horizontally and strikes a pendulum (

m_p=2.0\,\mathrm{kg}

l=1.5\,\mathrm{m}

) — perfectly inelastic collision. What is asked (solved below, a→e): (a) the disk's speed at the top of the plane; (b) the range and impact speed of the horizontal launch; (c) the speed after the inelastic collision with the pendulum and the energy lost; (d) the pendulum's maximum angle, the maximum tension, and whether it completes the loop; (e) the full energy balance (from spring to maximum angle).