Collisions and Momentum Conservation

A baseball player strikes a baseball with his bat sending the baseball flying off into the outfield. Homerun! In physics, the concept of inelastic collision describes the action of the bat hitting the baseball. The momentum of the baseball and the bat must be the same value before the collision as it is after the collision, which is why the ball flies off into the outfield after the collision. In this article, we will talk about different types of collisions and how they relate to the conservation of momentum and energy.

Collisions and Momentum Conservation Collisions and Momentum Conservation

Create learning materials about Collisions and Momentum Conservation with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Table of contents

    Meaning of Collisions and Momentum Conservation

    We see collisions occur all around us; a couple of examples are car collisions and the collision of a ball and a bat. In physics, when two objects experience a collision, each object exerts a very strong contact force on the other in a short amount of time. Generally, the forces that the objects exert on each other are very strong in comparison to any other forces acting on the system. Because of this, we can ignore the external forces and only consider the forces that the objects exert on each other. A good example of this is how in a car collision, the forces that cars exert on each other are much greater than the friction force from the road on the tires.

    A collision is an interaction between objects in which they exert very strong contact forces on each other over a short time period.

    Collisions and Momentum Conservation Car inelastic collision in which cars exert strong forces on each other and momentum is conserved StudySmarterCars in a collision exert strong contact forces on each other in a short time period, Pixabay

    The mechanical energy of a system, or the sum of the kinetic and potential energy, is conserved when only conservative forces are acting on objects in the system. If a nonconservative force, such as friction, acts on an object, the total mechanical energy is different than what it was initially. We categorize collisions as either elastic or inelastic based on whether or not the mechanical energy is conserved during the collision.

    An elastic collision is a collision in which the mechanical energy is conserved.

    An inelastic collision is a collision in which the mechanical energy is not conserved.

    Atoms and molecules experience elastic collisions frequently. The macroscopic collisions that we see every day are inelastic as kinetic energy converts into other forms of energy, like thermal energy, when macroscopic objects collide. We can, however, approximate that some macroscopic collisions are elastic, like collisions between billiard balls when playing pool. We can justify treating a collision as elastic only when the energy loss is negligible and there is virtually no deformation during the collision.

    Usually, during a collision, the objects in the collision get damaged from the large forces they experience. The forces experienced by the objects also vary in most collisions. If we use momentum conservation and energy conservation instead of Newton’s laws of motion to describe the collision, the problem becomes much simpler. We do not need to consider the varying force or how much damage is done to each object if we use the conservation of momentum.

    The Law of Conservation of Momentum

    When there are no external forces acting on a system, it is called an isolated system. When a collision occurs in an isolated system, momentum is conserved. The law of conservation of momentum tells us that momentum is constant before and after the collision. So, while mechanical energy is not always conserved during a collision, momentum in an isolated system is always conserved. As mentioned above, the forces exerted by the objects on each other are much greater than the external forces. So, for the examples we will consider, we will ignore any external forces and consider the system as isolated.

    The principle of momentum conservation states that in an isolated system, momentum is constant during a collision.

    Formulas for Collisions with Momentum Conservation

    Using the laws of conservation of momentum and energy, we can write down equations to describe the motion of objects in a collision. Here, we will focus on linear momentum, although this concept can also be applied in the form of angular momentum. Let’s consider separately the two types of collisions we mentioned above: elastic collisions and inelastic collisions.

    Formula for Elastic Collisions

    For elastic collisions, both mechanical energy and momentum are conserved. Thus we have an equation that we can write for each of these. The conservation of mechanical energy tells us that the energy before the collision,Ei, must equal the energy after the collision,Ef:

    Ei=Ef

    Collisions and Momentum Conservation Red striped billiard ball about to be hit by another billiard ball as an example of elastic collision in which momentum is conserved StudySmarterBilliard balls colliding is an example of an elastic collision, Pixabay

    The mechanical energy is the sum of the kinetic and potential energy of the system, but for now let’s consider a system that has just kinetic energy, like a billiard ball about to collide with another billiard ball. We will assume that both billiard balls are moving in a straight line and the collision is one-dimensional. The total energy before the collision is given by

    Ei=K1i+K2i =12m1v1i2+12m2v2i2,

    wherem1and m2are the masses of the billiard balls, andv1iandv2iare their initial velocities. Billiard balls typically have the same mass, but for generality we will assume they are slightly different. The total energy after the collision is given by

    Ef=K1f+K2f =12m1v1f2+12m2v2f2,

    wherev1fandv2fare the final velocities of the billiard balls. Equating these gives us:

    12m1v1i2+12m2v2i2=12m1v1f2+12m2v2f212m1v1i2+12m2v2i2=12m1v1f2+12m2v2f2m1v1i2+m2v2i2=m1v1f2+m2v2f2.

    This is the equation we have arrived at using the conservation of mechanical energy. Now, let’s look at the equation we can get from the conservation of momentum. The initial linear momentum,Pi, must equal the final linear momentum, Pf:

    Pi=Pf

    The linear momentum before the collision is

    Pi=P1i+P2i =m1v1i+m2v2i

    and the linear momentum after the collision is

    Pf=P1f+P2f =m1v1f+m2v2f.

    We get our equation from the conservation of linear momentum by equating these:

    m1v1i+m2v2i=m1v1f+m2v2f

    These equations look like they have a lot of unknown variables, but most problems give us the mass and initial velocities of the objects in the system. Recall that whenever we have two equations, we can solve for at most two unknown variables. Thus, we can then use both the conservation of energy and the conservation of momentum equations to solve for the final velocity of each of the two colliding objects.

    The equations for momentum and energy conservation used above only apply to objects that are traveling in one dimension and have an elastic collision. If the collision happens at an angle, we must consider the vertical and horizontal components of the velocities to find the components of the momentum.

    Formula for Inelastic Collisions

    For inelastic collisions, the mechanical energy is not conserved so we cannot equate the initial and final mechanical energies. We can, however, still use the conservation of momentum. Let’s consider a one-dimensional situation similar to the example above, except this time when the balls collide, some kinetic energy is lost to other forms of energy like thermal and sound energy. The equation to describe the conservation of momentum is the same as before:

    Pi=PfP1i+P2i=P1f+P2fm1v1i+m2v2i=m1v1f+m2v2f

    Since we only have one equation, we can only solve for one variable. We can do this if the problem gives us the values of the other variables, or we can solve for the ratio of the final velocities when given the values of the masses and initial velocities.

    A special case for inelastic collisions is when the objects stick together after the collision. This is called a perfectly inelastic collision. In this case, they share the same final velocity. Imagine that one of the balls from the example above is sticky and sticks to the other ball after the collision. Equating the initial momentum and the final momentum would then give us:

    Pi=PfP1i+P2i=P1,2fm1v1i+m2v2i=(m1+m2)vf

    For a perfectly inelastic collision, there is just one final velocity to find, which we can do if we are given the other variables.

    Other names for a perfectly inelastic collision are a completely inelastic collision, totally inelastic collision, maximally inelastic collision, etc. Different authors use different adjectives, but all these expressions mean the same thing.

    Collisions and Momentum Conservation Examples

    Since collisions happen frequently in our daily lives, we need to be able to set up energy and momentum conservation equations based on each situation. Below are a couple of examples to help you practice.

    A large block of massm1collides with a smaller block of massm2that is initially at rest. The blocks move together after the collision. What is the ratio of the initial and final velocities?

    Collisions and Momentum Conservation Example of two blocks of different masses experience a completely inelastic collision and move together after collision StudySmarterCompletely inelastic collision between blocks, StudySmarter Originals

    The collision between the blocks is a completely inelastic collision because kinetic energy is lost during the collision and the blocks move together after the collision. We will use the conservation of momentum to find the ratio of the velocities. The initial momentum of the larger block is

    P1i=m1vi.

    The smaller block is at rest before the collision and thus has zero initial momentum:

    P2i=0

    So, the total initial momentum is

    Pi=P1i+P2i=m1vi .

    We find the final momentum of the system by taking the combined mass multiplied by the final velocity so that

    Pf=(m1+m2)vf

    Equating these gives us:

    Pi=Pfm1vi=(m1+m2)vf

    Dividing the combined mass over to the left side and then the initial velocity to the right side gives us our ratio:

    vf=m1m1+m2vivfvi=m1m1+m2

    A billiard ball collides with another billiard ball that is initially moving with velocityv2i=1 ms. The two billiard balls go off at an angle ofθ=30°after the collision. The initial speed of the first billiard ball isv1i=4 ms, and they have equal masses. Find the final speeds of both billiard balls.

    Collisions and Momentum Conservation Collision between billiard balls in which they both go off at an angle after the collision StudySmarter

    Two-dimensional collision between billiard balls, StudySmarter Originals

    Let’s use the conservation of momentum to find the final velocities of the billiard balls. Since the billiard balls go off at an angle after the collision, we will need to split the initial and final momentum up into the vertical and horizontal components. We will start with the horizontal component. The horizontal component of the initial momentum is given by

    Pix=mv1i+mv2i

    There is no initial velocity in the vertical direction, so the initial momentum is zero:

    Piy=0

    To find the horizontal component of the final momentum, we need to split the velocity into vertical and horizontal components.

    The vertical components of each velocity are:

    v1fy=v1fsinθ

    v2fy=v2fsin(-θ) =-v2fsin(θ)

    . The horizontal components are:

    v1fx=v1fcosθ

    v2fx=v2fcos(-θ) =v2fcos(θ)

    Notice that we have a negative angle for the green billiard ball. The horizontal component for the final momentum is then:

    Pfx=mv1fx+mv2fx =m(v1fcosθ+v2fcosθ) ,

    and the vertical final momentum component is

    Pfy=mv1fy+mv2fy =m(v1fsinθ-v2fsinθ) .

    Now we can equate them to the components of the initial momentum:

    Pix=Pfxmv1i+mv2i=m(v1fcosθ+v2fcosθ)Piy=Pfy0=m(v1fsinθ-v2fsinθ)

    From the vertical component of the momentum, we can cancel the mass and the sine function:

    0=m(v1fsinθ-v2fsinθ)0=msinθ(v1f-v2f)

    Since we know that the scattering angle and the mass are not zero, we can divide them over to the other side to get:

    0msinθ=(v1f-v2f)0=(v1f-v2f)v1f=v2f

    If the scattering angle is zero, we would not be able to divide the sine term to the other side.

    We found that the magnitude of the velocity for the billiard balls is the same. So now let’s call itvf=v1f=v2f. We can use this in the equation for the horizontal momentum component to solve for the magnitude of the final velocity of both balls:

    mv1i+mv2i=m(v1fcosθ+v2fcosθ)m(v1i+v2i)=m(vfcosθ+vfcosθ)v1i+v2i=2vfcosθvf=v1i+v2i2cosθ=4 ms+1 ms2cos(30°)=2.89 ms

    The billiard balls both go off at an angle of30°with a velocity of2.89 ms.

    Collisions and Momentum Conservation - Key takeaways

    • In physics, a collision is an interaction between objects in which they exert very contact strong forces on each other in a short time period.
    • Momentum is always conserved when a collision occurs in an isolated system.
    • Mechanical energy is not always conserved during a collision.
    • An elastic collision is one in which the mechanical energy and momentum are conserved.
    • An inelastic collision is one in which the mechanical energy is not conserved, and there is a decrease in kinetic energy.
    Collisions and Momentum Conservation Collisions and Momentum Conservation
    Learn with 43 Collisions and Momentum Conservation flashcards in the free StudySmarter app

    We have 14,000 flashcards about Dynamic Landscapes.

    Sign up with Email

    Already have an account? Log in

    Frequently Asked Questions about Collisions and Momentum Conservation

    What is the meaning of collisions and momentum conservation?

    When there is a collision between objects in an isolated system in which they exert very strong forces on each other in a short time period, momentum is conserved.

    How do collisions with momentum conservation occur?

    A collision occurs when two objects exert strong forces on each other in a short time period. Momentum is conserved during the collision if the system is an isolated system.

    What is a good example of a collision with momentum conservation?

    An example of a collision in which momentum is conserved is when two billiard balls collide during a game of pool.

    What is the law of momentum conservation during collisions?

    The law of momentum conservation states that momentum is conserved during a collision in an isolated system in which there are no external forces acting on the system.

    How do you prove conservation of momentum in a collision?

    We can prove the conservation of momentum in a collision by finding the momentum of the system before and after the collision. The momentum in the system is constant and will be the same before and after the collision in an isolated system. 

    Test your knowledge with multiple choice flashcards

    In an elastic collision, momentum is conserved, but momentum is not conserved in an inelastic collision. 

    A stationary air track glider collides with a moving air track glider of the same mass. During the collision, the first glider loses all of its kinetic energy, and the second glider is set in motion at the same speed as the first glider. This is a completely inelastic collision since the first glider has lost all of its kinetic energy. 

    After the collision, if the objects stick together, the objects will have a common velocity. 

    Next

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Physics Teachers

    • 12 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner