Unit: Momentum
Chapter: Momentum and Impulse
Reference: AP Physics Algebra, Momentum, Momentum and Impulse, Concept of Momentum, Impulse, Collisions, Elastic and Inelastic Collisions, Collisions in One Dimension
After studying this chapter, you should be able to:
- State the concept of Momentum
- Know the meaning of Impulse and Collision
- Solve numerical related to collision
Concept of Momentum:
The product of mass m of a body and its velocity v is called its linear momentum p. Mathematically, we write
p = mv
In SI units, momentum is measured in kg ms–1. Momentum is a vector quantity. The direction of the momentum vector is the same as the direction of the velocity vector. The momentum of an object, therefore, can change on account of a change in its magnitude or direction or both. The following examples illustrate this point.
Impulse
The effect of a force applied for a short duration is called impulse. Impulse is defined as the product of force (F) and the time duration (Δt) for which the force is applied. i.e., Impulse = F.Δt
If the initial and final velocities of the body acted upon by a force F are u and v respectively then we can write
Impulse = mv
= mv – mu
= pf – pi
= Δp
That is, the impulse is equal to a change in linear momentum. Impulse in a vector quantity and its SI unit is kgms–1 (or N s).
Collisions
In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word collision refers to incidents in which two or more objects collide with great force, the scientific use of the term implies nothing about the magnitude of the force.
Some examples of physical interactions that scientists would consider collisions are the following:
- When an insect lands on a plant's leaf, its legs are said to collide with the leaf.
- When a boxer throws a punch, their fist is said to collide with the opponent’s body.
- When an astronomical object merges with a black hole, they are considered to collide.
Some colloquial uses of the word collision are the following:
- A traffic collision involves at least one automobile.
- A mid-air collision occurs between aeroplanes.
- A ship collision accurately involves at least two moving maritime vessels hitting each other; the related term, allision, describes when a moving ship strikes a stationary object (often, but not always, another ship).
Elastic and Inelastic Collisions
In an elastic collision:
1. Linear momentum of bodies before collision is equal to linear momentum of bodies after collision or linear momentum is conserved
2. Kinetic energy of the bodies before collision is equal to the kinetic energies of the interacting bodies after collision or kinetic energy is also conserved
So, if on the other hand the deformation may not be relieved and the two bodies could move together after the collision. A collision in which the two particles move together after the collision is called a completely inelastic collision.
The intermediate case where the deformation is partly relieved and some of the initial kinetic energy is lost is more common and is appropriately called an inelastic collision.
So, In an Inelastic Collision:
1. Linear momentum of bodies before the collision is equal to the momentum of bodies after a collision or linear momentum is conserved
2. Kinetic energy of the bodies before the collision is not equal to the kinetic energies of the interacting bodies after a collision or kinetic energy is not conserved
Collisions in One Dimension
Elastic collision in one dimension means considering collision between objects moving along a straight path so the path is along a straight line before and after the collision.
Do you think the collision of carom coins will be a collision in one dimension, also can we consider the carom coin collision in one dimension under any restricted condition?
Consider two bodies of masses m1 and m2 moving along a straight-line path in the same direction with velocities u1 and u2 respectively. The bodies collide elastically, and after collision let them start moving with velocities v1 and v2 along the same straight line.
Remember
· We are considering two body collision
· The two bodies move along a straight-line path so it is called ‘collision in one dimension’.
· Collision is an elastic collision
· We refer to 'before collision' and 'after collision' to state the condition of velocities, momentum, and kinetic energies before and after collision.
· Linear momentum is conserved
· Kinetic energy is conserved
Let us consider the total momentum of the colliding bodies before and after the collision.
Since the collision is elastic the two should be equal so we can write
𝒎𝟏𝒖𝟏 + 𝒎𝟐𝒖𝟐 = 𝒎𝟏𝒗𝟏 + 𝒎𝟐𝒗𝟐 ….. (1)
Because the collision is elastic the kinetic energy will also be conserved
From equation (1) we get
𝑚1 (𝑢1 − 𝑣1) = 𝑚2 (𝑣2 − 𝑢2) … … … … …. (3)
And from equation (2) we get
𝑚1 (𝑢12 − 𝑣12) = 𝑚2 (𝑣22 − 𝑢22) …………… (4)
Dividing equation (4) by (3) we get
𝒖𝟏 + 𝒗𝟏 = 𝒗𝟐 + 𝒖𝟐
Or 𝒖𝟏 − 𝒖𝟐 = 𝒗𝟐 − 𝒗𝟏…………………… (5)
or 𝒖𝟏 − 𝒖𝟐 = −(𝒗𝟏 − 𝒗𝟐) …………….. (6)
Equation (6) means: Relative Velocity before the collision is equal and opposite to the relative velocity of colliding particles/bodies after the collision.
Example 1:
Piter weighs 60 kg and travels with a velocity 1.0 m s–1 towards James who weighs 40 kg, and is moving with 1.5 m s–1 towards Piter. Calculate their momenta.
Solution: For Piter momentum = mass × velocity = (60 kg) × (1.0 m s–1) = 60 kgms–1
For James momentum = 40 kg × (– 1.5 ms–1) = – 60 kg ms–1
Note that the momenta of Piter and James have the same magnitude but they are in opposite directions.
Example 2: A rubber ball of mass 0.2 kg strikes a rigid wall with a speed of 10 ms–1 and rebounds along the original path with the same speed. Calculate the change in momentum of the ball.
Solution: Here the momentum of the ball has the same magnitude before and after the impact but there is a reversal in its direction. In each case, the magnitude of momentum is (0.2 kg) × (10 ms–1) i.e., 2 kgms–1.
If we choose the initial momentum vector to be along + x-axis, the final momentum vector will be along –x axis. So, pi = 2 kg ms–1, pf = –2 kgms–1. Therefore, the change in momentum of the ball, pf – pi = (–2 kgms–1) – (2 kgms–1) = – 4 kgms–1.
Here negative sign shows that the momentum of the ball changes by 4 kg ms–1 in the direction of the x-axis. What causes this change in the momentum of the ball? In actual practice, a rubber ball rebounds from a rigid wall with a speed less than its speed before the impact. In such a case also, the magnitude of the momentum will change.
Key Points:
- Momentum and impulse are two related concepts in physics that describe the motion of an object.
- Momentum is a property of an object that is related to its velocity and mass. It is defined as the product of an object's mass and velocity: p = mv, where p is momentum, m is mass, and v is velocity. Momentum is a vector quantity, meaning it has both magnitude and direction.
- An impulse is a change in momentum that occurs when a force is applied to an object for a period of time. It is defined as the product of the force applied and the time for which it is applied: J = Ft, where J is impulse, F is force, and t is time. Impulse is also a vector quantity.
- The relationship between momentum and impulse is described by the impulse-momentum theorem, which states that the impulse on an object is equal to the change in momentum of the object: J = Δp.
- This theorem can be used to analyze the motion of objects in collisions and other situations where forces are applied for a period of time. For example, in a car crash, the force of the collision applies an impulse to the car, causing it to change momentum and come to a stop. The impulse-momentum theorem can be used to calculate the force of the collision and the change in momentum of the car.
In physics, impulse and collision are related concepts that are used to describe the behaviour of objects when they come into contact with each other.
An impulse is a force that acts on an object over a short period of time. It is defined as the change in momentum of an object, and it is equal to the force multiplied by the time during which it acts. Impulse is a vector quantity and is usually represented by the symbol J.
A collision is an event in which two or more objects come into contact with each other. During a collision, the objects may exchange energy, momentum, or both. Collisions can be classified into two types: elastic collisions and inelastic collisions. In an elastic collision, both momentum and kinetic energy are conserved, while in an inelastic collision, only momentum is conserved.
The concept of impulse is particularly useful in analyzing collisions, as it allows us to calculate the change in momentum of an object during a collision. This, in turn, allows us to calculate the force that was exerted on the object during the collision.
Collision key points, on the other hand, may refer to specific points on objects that come into contact during a collision. These key points can be used to determine the orientation and position of the objects after the collision and can be used to calculate the impulse and force involved in the collision.