Impulse, momentum and collisions (Topic 6A)
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The net force on an object is defined as the change in momentum over a period of time:
Where:
- is the net force in ,
- is the change in momentum in and
- is the change in time in .
Newton’s Second Law is a specific case of net force where the object’s mass remains constant. However, what if the mass of an object with a net force applied to it changes with time? Then, only the general form is applicable:
Rockets ejecting fuel as they accelerate are a good example of this. The rocket loses mass as it accelerates.
At time the rocket has mass , but after a period of time , fuel of mass has been ejected from the rocket, making it less massive. The net force on the rocket has remained constant during this time:
Since the change in momentum remains constant over time, the rocket’s velocity must increase with time as its mass decreases.
Question walkthrough
Finding Mass Using Force and Momentum Change
Calculates a car's mass from driving force and change in velocity over time using F = dp/dt, without first finding acceleration.
Impulse is the effect of a force acting on an object over a given time period :
Where:
- is the impulse in ,
- is the force in and
- is the change in time in .
Impulse can be considered the change in momentum of an object.
A change in momentum is a result of a change in velocity at constant mass:
Where:
- is the impulse in ,
- is change in momentum in ,
- is mass in
- and is the change in velocity in .
An excellent illustration of impulse is rain versus hail when standing under an umbrella. Raindrops are smaller and have less mass, and when they hit an umbrella, they have a relatively small change in momentum.
On the other hand, hail has a larger mass, and when it hits an umbrella, there is a greater change in momentum and, therefore, impulse. Standing in hail with an umbrella, a person will feel significantly more force per second than in rain.
Question walkthrough
Impulse Delivered to a Returning Tennis Ball
Calculate the impulse and direction of the force delivered by a racket to a tennis ball that rebounds at a lower speed.
The principle of conservation of momentum states that the total momentum of a closed system remains constant. A closed system is one in which no external forces act upon it.
A system can consist of many objects that interact with each other:
- The objects can interact through contact forces, like snooker balls bouncing off each other.
- Objects can also interact through non-contact forces, such as the electrostatic repulsion between two electrons or the gravitational attraction between planets.
The laws of classical mechanics dictate that momentum is always conserved. This means that the total momentum of all interacting bodies before and after a collision is the same.
The conservation of momentum can be used to calculate the velocity of objects before and after a collision. For instance, two objects colliding as illustrated below:
The principle of conservation of momentum in a closed system can be stated mathematically as:
Momentum is a vector quantity. Therefore, both the magnitude and direction of the momentum vector are conserved for a closed system.
In one dimension, the direction of the momentum vector does not change after an interaction, since the objects move along one direction. For example, consider the head-on collision of Ball A and Ball B of equal mass :
The momentum of an object with mass moving at speed is equal to:
Therefore, the principle of conservation of momentum for this collision leads to the expression:
Where is the final speed of Ball A. Cancelling the masses and rearranging gives:
If a collision between two objects occurs at an angle in two-dimensions, their final velocities will be in different directions.
Ball A moves at to the right, and collides at an angle with a stationary Ball B of equal mass; they move off in directions to each other.
Since momentum is conserved, the momentum vectors of the balls after the collision must sum to the momentum vector of Ball A before the collision.
The masses are the same, so the velocity vector triangle is the same shape as the momentum vector triangle. The vector triangle is a right-angle triangle, so the final velocity of Ball B can be found from Pythagoras’ theorem:
Therefore:
Conservation of momentum requires that momentum is conserved in any direction.
The diagram illustrates a collision between Ball A of mass and initial velocity along the direction, colliding with Ball B of mass The collision results in final velocities and at angles of and to the horizontal.
Conservation of momentum in the direction leads to:
Since there is no initial momentum in the direction, the expression for the direction is:
In a collision, both the total momentum and the total energy are conserved. However, the kinetic energy before the collision can be converted to other forms of energy, such as heat or sound.
- Perfectly elastic collisions: both the total momentum and the total kinetic energy are conserved.
- None of the kinetic energy is converted to other forms.
- Inelastic collisions: the total momentum is conserved, but the total kinetic energy is not.
- Some of the kinetic energy is converted to other forms.
Question walkthrough
Proving a Collision is Elastic
Elastic collision
Question walkthrough
Classifying a Collision as Elastic or Inelastic
Determining collision type
Momentum describes how much motion an object has and its resistance to change in velocity:
Where:
- is the momentum in ,
- is mass in , and
- is velocity in .
Momentum is proportional to velocity. Velocity is a vector with a magnitude component (speed) and a direction. Hence, momentum is also a vector with both magnitude and direction.
Scalar quantities only have magnitude, such as mass and speed, and therefore do not change with direction.
Kinetic energy is the energy possessed by an object due to its motion. Kinetic energy is a scalar quantity that has a magnitude but no direction. The formula that describes the kinetic energy is:
Where:
- is the mass in kilograms (), and
- is the speed in metres per second ().
The kinetic energy of an object is measured in joules () and is directly proportional to its mass and to the square of its speed.
For example, doubling the mass of an object moving at the same speed would double its kinetic energy. Alternatively, doubling the velocity of the same object would quadruple its kinetic energy.
The kinetic energy of a non-relativistic particle can be expressed in terms of its momentum rather than its velocity:
Where:
- is the kinetic energy in ,
- is the momentum in , and
- is the mass in .
Physically, this form says that, for a given mass, a particle’s kinetic energy grows with the square of its momentum. It is particularly useful in further mechanics and particle physics problems where momentum is the natural variable (e.g. charged particles in magnetic fields), so kinetic energy can be obtained without first calculating velocity.
The equation is only valid for non-relativistic particles, those moving at speeds well below the speed of light, c. Particles approaching gain significant relativistic mass, which breaks the above equation. A common rule of thumb is that the non-relativistic form is acceptable while .
The derivation of the kinetic energy equation for a non-relativistic particle in terms of momentum combines the two equations already known from earlier in the specification: the definition of momentum and the kinetic energy of a body.
Firstly, the kinetic energy of a body is:
While the definition in momentum with velocity as the subject is:
Now, subtitute the expression for into the kinetic energy equation:
Finally, cancel one factor of to arrive at the correct equation: