ChemPhys 173/273

Unit 12: Momentum and Collisions

Problem Set B

Overview:

Problem Set B targets your understanding of momentum conservation and your ability to apply it to the analysis of a collision or explosion.

The Momentum Conservation Principle

In a collision between two objects, each object is interacting with the other object. The interaction involves a force acting between the objects for some amount of time. This force and time constitutes an impulse and the impulse changes the momentum of each object. Such a collision is governed by Newton's laws of motion; and as such, the laws of motion can be applied to the analysis of the collision (or explosion) situation. So with confidence it can be stated that ...

In a collision between object 1 and object 2, the force exerted on object 1 (F1) is equal in magnitude and opposite in direction to the force exerted on object 2 (F2). In equation form:

### F1 = - F2

The above statement is simply an application of Newton's third law of motion to the collision between objects 1 and 2. Now in any given interaction, the forces which are exerted upon an object act for the same amount of time. You can't touch another object and not be touched yourself (by that object). And the duration of time at which you act upon that object is the same as the duration of time at which that object acts upon you. Touch the wall for 2.0 seconds, and the wall touches you for 2.0 seconds. Such a contact interaction is mutual; you touch the wall and the wall touches you. It's a two-way interaction - a mutual interaction; not a one-way interaction. Thus, it is simply logical to state that in a collision between object 1 and object 2, the time during which the force acts upon object 1 (t1) is equal to the time during which the force acts upon object 2 (t2). In equation form:

### t1 = t2

The basis for the above statement is simply logic. Now we have two equations which relate the forces exerted upon individual objects involved in a collision and the times over which these forces occur. It is accepted mathematical logic to state the following:

If A = - B

and C = D

then A • C = - B • D

The above logic is fundamental to mathematics and can be used here to analyze our collision.

If F1 = - F2

and t1 = t2

then F1 • t1 = - F2 • t2

The above equation states that in a collision between object 1 and object 2, the impulse experienced by object 1 (F1 • t1) is equal in magnitude and opposite in direction to the impulse experienced by object 2 (F2 • t2). Objects encountering impulses in collisions will experience a momentum change. The momentum change is equal to the impulse. Thus, if the impulse encountered by object 1 is equal in magnitude and opposite in direction to the impulse experienced by object 2, then the same can be said of the two objects' momentum changes. The momentum change experienced by object 1 (m1v1) is equal in magnitude and opposite in direction to the momentum change experienced by object 2 (m2v2). This statement could be written in equation form as

m1 v1 = - m2 v2

This equation claims that in a collision, one object gains momentum and the other object loses momentum. The amount of momentum gained by one object is equal to the amount of momentum lost by the other object. The total amount of momentum possessed by the two objects does not change. Momentum is simply transferred from one object to the other object.

The logic above was used to provide an argument for that fact that in any collision between object 1 and object 2:

• the forces encountered by the objects are equal in magnitude and opposite in direction
• the times over which the interaction occurs is the same for each object
• the impulses delivered to each object are equal in magnitude and opposite in direction
• the momentum changes of each object are equal in magnitude and opposite in direction

To be successful on this problem set, you will have to:

• use the momentum change equation ( m1v1 = - m2v2 ) with comfort and confidence.
• be cognizant of units and perform unit conversions where necessary.
• recognize that a change in a quantity is calculated as the final value of the quantity minus the initial value of the quantity.
• utilize the kinematic equation d = vo•t + 0.5•a•t2 equation in frictionless situations in which the a term cancels.
• utilize energy conservation principles to determine the height to which a pendulum rises
• practice the habits of a good problem solver.

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

Momentum Conservation Principle | Isolated Systems | Collisions | Explosions

View Sample Problem Set.

 Problem Description Audio Link 1 Determination of the recoil speed of a child who has thrown a stone forward. 2 Determination of the recoil speed of a rifle after firing a bullet. 3 Determination of the recoil speed of a rifle/man after firing a bullet. 4 Detrmination of the ratio of post-explosion speeds of two popcorn kernel fragments if given their mass ratios. 5 Determine the post-explosion speed of a male skater who propels a female skater forward from an orignally moving state. 6 Determination of the time required for a man to move from the middle of a pond (on ice) to the edge after imparting an impulse to a physics textbook. 7 Analysis of the collision between a bowling ball and a bowling pin to determine the post-collision speed of the bowling ball. 8 Determine the post-impulse speed of a skater who propels a snowball forward from an orignally moving state. 9 Determination of the post-impulse speed of a boat after a swimmer dives off it. 10 Two-step analysis of an astronaut who hurls a wrench in space to propel him/herself backwards; time to travel a given distance must be determined. 11 Momentum plus problem in which a bird propels itself off a swing, setting the swing into backward motion; must determine the height to which the swing swings. 12 Analysis of an explosion to determine post-explosion velocity of cannon and cart. 13 Analysis of an explosion to determine post-explosion velocity of cannon and cart; cart and ball are originally in motion. 14 Determine the post-explosion speed of a skater who propels a snowball forward from an orignally moving state. 15 Analysis of a collision between a moving cart and a dropped brick in order to determine the mass of the dropped brick.

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