Complete Theory

Worked Examples

Example 1Elastic 1D collision — equal mass billiard balls

Example 2Perfectly inelastic collision — vehicle crash

Example 3Oblique billiard collision: what the line of centres is

Example 4Oblique collision with unequal masses (normal/tangential decomposition)

Exercises with Solutions

Exercise 1Elastic 2D collisionHard

📋 Problem to solve

Two balls of equal mass

m_1 = m_2 = 0.3\,\mathrm{kg}

collide elastically. Ball 1 moves initially with speed

v_1 = 6\,\mathrm{m/s}

east, while ball 2 is at rest. After the collision, ball 1 is deflected by

\theta_1 = 30°

from its original direction. Determine: (a) the speed

v_1'

of ball 1 after the collision, (b) the speed

v_2'

and direction

\theta_2

of ball 2 after the collision.

📌 Given data

m_1 = m_2 = 0.3\,kg (equal masses)v_1 = 6\,m/s (initial speed of ball 1)v_2 = 0 (ball 2 at rest)\theta_1 = 30° (deflection angle of ball 1)

Exercise 2Rolling diskVery Hard

📋 Problem to solve

A uniform solid disk of mass

M = 2\,\mathrm{kg}

and radius

R = 0.15\,\mathrm{m}

rolls without slipping down an incline of

\theta = 25°

and length

L = 3\,\mathrm{m}

, starting from rest. Determine: (a) the acceleration of the centre of mass, (b) the CM speed at the bottom of the incline, (c) the static friction force required for rolling.

📌 Given data

M = 2\,kg (disk mass)R = 0.15\,m (disk radius)\theta = 25° (incline angle)L = 3\,m (incline length)

📚

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).