Exercises on collisions and momentum

Welcome to the collection of physics exercises focused on collisions and momentum, designed to provide you with a wide range of problems from fundamental concepts to more complex scenarios, thus helping to consolidate your understanding of the subject.

The collisions represent one of the most fascinating and studied phenomena in physics because they involve such fundamental principles as conservation of momentum and energy.

Basically, collisions fall into two categories: collisions elastic, where both momentum and the total kinetic energy of the system are conserved, and serve as a useful model for understanding fundamental interactions between particles; collisions inelastic, where although the total momentum of the system is conserved, some of the kinetic energy is converted into other forms of energy, such as heat or internal energy. This type of collision is typical in many practical applications, such as automobile accidents.

Each exercise is accompanied by detailed solutions to facilitate learning and ensure that you can check your answers.

For errors or typos, contact us here: info@gohpysics.it

Still struggling with the topic? If you need to brush up on the topic, book a session at tutoring with our tutors!

Exercises on collisions and momentum

1

On the fully inelastic collision

A sphere of mass $m = 10 \ kg$ moves with initial velocity $v = 1.5 \ m/s$ . If a second sphere of mass $M = 90 \ kg$ and initial speed $V = 9.8 \ m/s$ bumps into the smaller sphere, and after the bump the two spheres remain attached

two spheres moving with initial velocity

Calculate the final velocity $v_f$ of the two spheres

[ $v_f = 9.0 \ m/s$ ]

2

On the elastic impact of two objects moving toward each other

A yellow ball of mass $M = 1.2 \ kg$ and initial speed $V_i = 1.5 \ m/s$ elastically bumps against a red ball of mass $m = 0.7 \ kg$ and velocity in modulus $v_i = 0.8 \ m/s$ Moving toward the yellow marble, as shown in the figure

two spheres moving into each other

Calculate final velocities $V_f$ e $v_f$ Of the yellow marble and the red marble, respectively.

[ $V_f = -0.2 \ m/s\, ; \, v_f = 2.1 \ m/s$ ]

3

On the elastic impact of a moving object and a stationary object with similar masses

A yellow ball of mass $M = 1.2 \ kg$ and initial speed $V_i = 1.5 \ m/s$ elastically bumps against a red ball of mass $m = 0.7 \ kg$ initially stationary, as shown in Figure

one sphere moving into another static sphere

Calculate final velocities $V_f$ e $v_f$ Of the yellow marble and the red marble, respectively.

[ $V_f = 0.4 \ m/s \, ; \, v_f = 1.9 \ m/s$ ]

4

On the elastic collision of a moving and a stationary object with very different masses-Case 1

A yellow ball of mass $M = 1.2 \ kg$ and initial speed $V_i = 1.5 \ m/s$ elastically bumps against a red ball of mass $m = 99 \ kg$ initially stationary, as shown in Figure

elastic collision diagram between two spheres, one way heavier than the other

Calculate final velocities $V_f$ e $v_f$ Of the yellow marble and the red marble, respectively.

[ $V_f = -1.5 \ m/s \, ; \, v_f = 0 \ m/s$ ]

5

On the elastic collision of a moving and a stationary object with very different masses-Case 2

A yellow ball of mass $M = 99 \ kg$ and initial speed $V_i = 1.5 \ m/s$ elastically bumps against a red ball of mass $m = 1.2 \ kg$ initially stationary, as shown in Figure

elastic collision diagram between two spheres

Calculate final velocities $V_f$ e $v_f$ Of the yellow marble and the red marble, respectively.

[ $V_f = 1.5 \ m/s \, ; \, v_f = 3.0 \ m/s$ ]

6

On the elastic impact of a moving object and a stationary object with equal masses

A yellow ball of mass $M =1.2 \ kg$ and initial speed $V_i = 1.5 \ m/s$ elastically bumps against a red ball of mass $m = M = 1.2 \ kg$ initially stationary, as shown in Figure

elastic collision diagram between two spheres, one starting with zero velocity

Calculate final velocities $V_f$ e $v_f$ Of the yellow marble and the red marble, respectively.

[ $V_f = 0 \ m/s \, ; \, v_f = 1.5 \ m/s$ ]

7

On elastic collision in two dimensions

A red ball of mass $m = 1.5 \ kg$ and speed $v_i = 2.1 \ m/s$ bumps into an initially stationary green marble ( $V_i = 0$ ) and mass $M = 3.2 \ kg$ . Since the collision is not head-on, after the collision the red marble continues at speed $v_f = 0.7 \ m/s$ In the direction shown in the figure, at an angle $20^\circ$ with respect to the initial direction. Instead, the green marble will move in a direction that forms an angle equal to $\theta$ with respect to the initial direction

elastic collision diagram between two spheres at an angle

Calculate the final velocity $V_f$ of the green marble
Calculate the angle $\theta$

[Hint: Assume the impact is elastic]

[ $V_f = 1.4 \ m/s \, ; \, \theta = 61^\circ$ ]

8

On the ballistic pendulum and inelastic impact

A mass case $M = 3.0 \ kg$ hangs from an ideal wire connected to the ceiling. A projectile of mass $m = 0.5 \ kg$ is fired at a speed $v_i = 1.2 \ m/s$ and sticks inside the crate. If after the impact, the crate+projectile system possesses a velocity $v_f$

a projectile and a pendulum in an inelastic collision

Calculate the maximum height $h_f$ achieved by the cash+projectile system

[ $h_f = 1.5 \cdot 10^{-3} \ m$ ]

9

On the spring and inelastic collision

A projectile of mass $m = 2.5 \ kg$ and speed $v_i = 1.5 \ m/s$ makes a totally inelastic impact with a mass case $M = 1.0 \ kg$ initially stationary. The case is attached to a spring of elastic constant $k$ . If the maximum compression of the spring is $\Delta x = 2.5 \ cm$

inelastic collision between a bullet and a box connected to a spring

Calculate the spring's elastic constant $k$

[ $k = 6.4 \cdot 10^3 \ N/m$ ]

10

On inelastic collision in two dimensions.

A red ball of mass $m = 1.0 \ kg$ and speed $v = 1.4 \ m/s$ collides against a green ball of mass $M = m = 1.0 \ kg$ and speed $V = 0.8 \ m/s$ . The directions of motion of the two balls are perpendicular to each other, as shown in the figure. If the collision is completely inelastic

elastic collision between two spheres moving orthogonally

Calculate the final velocity $v_f$

[ $v_f = 0.8 \ m/s$ ]

11

On inelastic collision and range of motion.

Consider the following figure in which a blue ball of mass $M = 10 \ kg$ initially stationary is attached to a spring of elastic constant $k = 96 \ N/m$ and tablet of $\Delta x = 0.5 \ m$ . At a certain instant the spring is released and the blue ball of mass $M$ totally inelastically impacts a second red ball of mass $m = 5.0 \ kg$ . Both balls are initially located at a height $h = 8.6 \ m$

collision between two spheres and one then falling from a height

Calculate the distance $s$ Where the system touches the ground

[ $s = 1.3 \ m$ ]

12

On the totally inelastic collision and the conservation of angular momentum - Case 1

A full and homogeneous disk of mass $M = 2.5 \ kg$ and radius $R = 1.0 \ m$ Rotates in a horizontal plane relative to an axis passing through the center with constant angular velocity $\omega_i = 2.5 \ rad/s$ . At a certain instant a ball of mass $m = 0.2 \ kg$ bumps totally inelastically on the edge of the disk hitting it at a speed $v_i = 4.2 \ m/s$ , as shown in the figure

conservation of angular momentum in a collision

Calculate the angular velocity $\omega_f$ Of the system after the collision

[ $\omega_f = 2.2 \ rad/s$ ]

13

On totally inelastic collision and conservation of angular momentum - Case 2

A full and homogeneous disk of mass $M = 2.5 \ kg$ and radius $R = 1.0 \ m$ free to rotate in a vertical plane with respect to an axis passing through the center is initially stationary. At a certain instant a small ball of mass $m = 0.2 \ kg$ bumps totally inelastically on the edge of the disk hitting it at a speed $v_i = 4.2 \ m/s$ , as shown in the figure

Calculate the angular velocity $\omega_f$ Of the system after the collision

[ $\omega_f = 0.6 \ rad/s$ ]

14

On the totally inelastic collision

The mass box $M = 5.8 \ kg$ shown in the figure contains within it a small wooden case of mass $m = 1.3 \ kg$ initially stationary. At a certain instant, a projectile of mass $m_p = 0.5 \ kg$ and speed $v = 8.5 \ m/s$ Totally inelastically impacts the initially stationary box

Calculate the final velocity $V_f$ of the mass box $M$ and the final speed $v_f$ of the mass case $m$

[ $v_f = 0 \ m/s \, ; \, V_f = 0.7 \ m/s$ ]

15

On the elastic impact with a bar

A bar of length $L = 5.0 \ m$ and mass $M = 10 \ kg$ is constrained to rotate in the vertical plane around a pivot placed at either end, as shown in the figure. A ball of mass $m = 2.5 \ kg$ elastically impacts with velocity $v_i = 9.0 \ m/s$ the bar and bounces with speed $v_f = 8.5 \ m/s$

Calculate the angular velocity $\omega$ of the bar

[ $\omega = 0.6 \ rad/s$ ]

Useful Links

collisions Momentum