Momentum and Impulse

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this product is intended for the serious physics student, if you are not a serious physics student,please use version 1.3, momentum and impulse for dummies

Version: 1.29

Introduction to Momentum

• Momentum is the “quantity of motion.”

• Momentum is represented by “p”.

• Momentum is the product of the mass and the velocity of an object or: p=mv

• Momentum is a vector quantity, meaning it has both magnitude and direction.

• Measured in units of kg m/s

Introduction to Impulse• Impulse is the product of the net force acting on a

given body during the time it acts. Units are N s.• Impulse = F(Δt) • Impulse = Change in Momentum

• “A large change in momentum occurs only when there is a large impulse. A large impulse, however, can result from either a large force acting over a short time, or a smaller force acting over a longer time.”

F * t = F * T

F t p f iF t mv mv

Remember Newton’s 2nd Law?

.. ?

F ma

vF m

tFt m v

Look familiar

Conservation of Momentum• The total momentum of a system of objects

must remain constant unless outside forces act upon the system.

• Momentum, like other physical quantities, cannot be destroyed, merely redistributed.

• Momentum is distributed through impulse, or the change in momentum (see how this all fits together yet?)

• Most studies of the conservation of momentum involve collisions.

Types of Collisions• Perfectly (completely) Inelastic Collisions -collisions where

the objects stick together and move as one. The mass after the collision is equal to the sum of the masses and there is just one final velocity. Kinetic energy is not conserved since a lot of kinetic energy is converted to heat or other energies.

• Perfectly (completely) Elastic Collisions – collisions where the objects bounce off of each other or separate. They then return to their original shape. Kinetic energy is conserved.

• Many collisions are neither completely elastic or inelastic. They separate after the collision but do not return to their original shape, so heat is created and kinetic energy is lost.

• Separation or Explosion Collisions – the objects are at rest initially or are all one object and then they separate.

Inelastic - Collisions that Stick

• For Collisions that Stick, the after mass is the sum of all the masses, and there is a common or shared velocity.

• Remember:

pinitial = pfinal

• m1v1+m2v2= (m1+m2) v’

m1

(m1 + m2)

m2

v1 v2

V’

Elastic Collisions

• Remember: pinitial = pfinal

• m1v1+m2v2 =m1v1’+m2v2’

• Usually mass remains the same after the collision but this does not always happen.

• ’ means after the collision

m1 m2

Before

After

v1

v1’ v2’m1 m2

Perfectly Elastic Collisions• Objects separate after the collision and

return to their original size and shape

• Momentum is conserved.

• Kinetic energy is also conserved.

'' 'A A B B A A B Bm v m v m v m v

2 2 2 21 1 1 1' '

2 2 2 2A A B B A A B Bm v m v m v m v

Separations/ExplosionsSometimes objects start as one and separate into two or more parts moving with individual velocities.

– Example: A bullet is fired from a rifle. The bullet leaves the rifle with a velocity of vb and as a result the rifle has a recoil velocity of vr.

– Solution: Before the rifle is fired the momentum of the system is zero. So…

0 = mbvb’ + mrvr’Note: for total momentum to remain zero, the

rifle’s momentum must be equal to but in the opposite direction to the bullet’s momentum

(mrvr’ = - mbvb’)

The Law of Momentum Conservation

Momentum Conservation Principle

• For a collision occurring between object 1 and object 2 in a closed, isolated system, the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision. That is, the momentum lost by object 1 is equal to the momentum gained by object 2.

A closed isolated systemis a set of objects which encounter no forces or influences otherthan those they exert on each other.

• Consider a collision between two objects - object 1 and object 2. For such a collision, the forces acting between the two objects are equal in magnitude and opposite in direction (Newton’s Third Law)

• Example: The force of a baseball bat on a ball is equal but opposite the force of the ball on the bat.

• The forces act between the two objects for a given amount of time.

• Regardless of how long the time is, it can be said that the time that the force acts upon object 1 is equal to the time that the force acts upon object 2.

• Example: the time that the force of the bat acts on the ball is equal to the time the force of the ball acts on the bat.

• Since the forces between the two objects are equal in magnitude and opposite in direction, and since the times for which these forces act are equal in magnitude, it follows that the impulses experienced by the two objects are also equal in magnitude and opposite in direction.

• Example: The impulse acting on the bat is equal but opposite the impulse acting on the ball (since F and t are the same).

• Since each object experiences equal and opposite impulses, it follows logically that they must also experience equal and opposite momentum changes.

• Example: The change in momentum is the same for the ball and the bat (since the impulses are the same: Impulse = change in momentum).

• In a collision, the momentum change of object 1 is equal and opposite to the momentum change of object 2. That is, the momentum lost by object 1 is equal to the momentum gained by object 2.

Why use an Airbag?

• Since the change in momentum and thus the impulse stays the same whether or not you have an airbag, the force with which you hit will be decreased by increasing the time over which the force acts.

Δp = F t

m (vf – vi) = F t

m (vf - vi ) = F t

Why follow through in your golf swing?

• By following through you are increasing the time that the club is in contact with the ball. Thus you are increasing the impulse (assuming force is the same) which results in a greater change in momentum. That means a greater acceleration or change in velocity so the ball will go faster and/or farther (if a tree or other obstacle doesn’t get in its way!).

F t = Δ p

F t = m Δv

F t = m Δv

• Consider a collision in football between a fullback and a linebacker during a goal-line stand. The fullback plunges across the goal line and collides in midair with linebacker. The linebacker and fullback hold each other and travel together after the collision. The fullback possesses a momentum of 100 kg*m/s, East before the collision and the linebacker possesses a momentum of 120 kg*m/s, West before the collision. The total momentum of the system before the collision is 20 kg*m/s, West.

• Therefore, the total momentum of the system after the collision must also be 20 kg*m/s, West. The fullback and the linebacker move together as a single unit after the collision with a combined momentum of 20 kg*m/s. Momentum is conserved in the collision.