Optional Unit V: Applications of Kinematics and Dynamics
A. Momentum

1. Impulse and Momentum

Key Concepts

The momentum of a moving object is expressed as:

The units for momentum are kgùm/s. (Nùs, the units for impulse, can be used interchangeably).

Momentum is a vector quantity. Its direction is always the same as the velocity.

Impulse is a vector quantity acting in the same direction as the force which caused the impulse.

The units for impulse are Nùs.

The area under a force versus time graph can be used to determine the impulse (provided that the direction of the force producing the impulse is not changing).

An impulse acting on an object results in a change in momentum, .

The impulse of a force is:

Learning Outcomes

Students will increase their abilities to:

1. Define the following terms: momentum, impulse.

2. Use the correct SI units for impulse momentum.

3. Apply an understanding of vectors to momentum.

4. Compare the directions of the momentum, impulse, force, and velocity vectors in a given situation.

5. Recognize that an impulse acting on an object results in a change in momentum.

6. State the relationship between the impulse of a force acting on an object and its change in momentum.

Teaching Suggestions, Activities and Demonstrations

1. Perform one or more activities to investigate the Law of Conservation of Momentum in one or two dimensions.

2. Load two spring-loadable dynamics carts end to end on a flat surface. Place different masses on each cart. Attach a recording timer to each cart. Start the recording timers. Press the trip release on the dynamics carts. Determine the velocity of each cart after the "explosion." Find the mass of each cart. Determine the momentum of each cart. Compare the total momentum before and after the "explosion."

3. Develop a force versus time graph to determine the impulse acting in some given situation.

4. Have two students standing on skateboards face each other. One student throws a medicine ball (or a basketball) to the other student. Qualitatively describe what happens. Repeat several times with students of different masses. As a variation on this, have the two students standing close together and push off from one another so that they are propelled in opposite directions. One student pushes off, and the other student simply braces himself or herself. Repeat, reversing which student is pushing off. Ask the students to describe what happens, and to determine if it matters which of the two students did the pushing.

   Use helmets, elbow pads, knee pads and other protective clothing when using skateboards.

5. Attach a dynamics cart (or a skateboard) to a recording timer on a flat surface. Set the cart in motion. Carefully drop a brick on top of the moving cart. This might have to be practised a few times, so that the brick lands on the cart and does not fall off. Find the change in the horizontal and vertical momentum of the brick. Find the change in the horizontal momentum of the cart. Determine if momentum is conserved.

   Change the mass of the dynamics cart. Repeat the experiment. Predict the final velocity of the cart. Check the prediction against the experimental results.

6. Using dry ice pucks, a linear air track, or an air table, qualitatively investigate collisions between objects of different mass. (See if you can get permission from a local curling or hockey rink to design and perform some experiments on a flat ice surface. Students could work in groups on different parts of the ice surface. The physics of curling would be a fascinating topic to study.)

   Design a variety of trials to demonstrate elastic and inelastic collisions. Have the students develop some method for analyzing the results (e.g., video recording, strobe photography, slow motion photography, recording timers, stopwatches and carbonless paper marking tracks, digitized image processing, ink tracks, or whatever). Using vector analysis, examine the collisions in detail, searching for patterns and generalizations that can be made. This kind of open-ended activity is excellent for developing Critical and Creative Thinking, and Independent Learning, has some interesting possibilities, and for some activities is to be preferred to the use of "cookbook approaches" for experimentation.

7. To construct a fairly inexpensive ballistic pendulum, take a cinder block or brick and attach it with ropes to a sturdy ceiling support. A flow pen can be used to mark a vertical reference line on the side of the block. Tie on some modelling clay with string to the side of the block. A heavy pendulum can be attached from the ceiling close to the ballistic pendulum, so that as the pendulum swings it will strike the block. A pellet rifle, clamped securely, can be used to fire pellets into the modelling clay on the side of the block. (Safety glasses should be worn.)

   (Caution: Any activities which propel arrows or other high velocity projectiles at a ballistic pendulum are recommended only under extremely well controlled teacher supervision and controlled experimental conditions.)

8. Have the students bring in a variety of different types of balls to class. Design an experiment to determine the coefficient of restitution for the balls. One way of doing this is to use a meter stick as a reference scale. Drop a ball from a fixed height, such as one meter, and observe how far up the scale it reaches. This gives a way of determining the coefficient of restitution. The ratio of velocities from and to the rebounding surface is related to the ratio of final and initial displacements if the balls are projected vertically. If a ball has plenty of bounce, the coefficient of restitution approaches 1, while a ball with little bounce has a coefficient closer to 0.

   Some interesting types of questions can be explored, such as: What happens if the ball is released from a different initial height? Is the ratio of final to initial displacements constant? How does this ratio vary from one type of ball to another? How is this ratio affected when the air pressure inside a basketball is allowed to vary? For a given type of ball, such as tennis balls or golf balls, do some brands bounce better than others? The students will likely think of other types of questions to ask. They should develop a hypothesis, and design and perform experiments to test the hypothesis.

9. Have the students bring in a variety of spinning tops. They can qualitatively examine the tops, identifying common characteristics such as shape and distribution of mass, location of the centre of gravity, and so on. Moments of inertia, conservation of angular momentum, precession, and angular acceleration can also be investigated at a more advanced level.

10. Take a bicycle wheel (removing the front wheel is relatively easy) and hold it vertically by the axle on either side. Spin the wheel. Try to turn the wheel so that the axle tilts vertically. Notice the gyroscopic effect. Use other gyroscopes to illustrate what is happening .

    As a variation on the same theme, mount one end of the bicycle axle (the hub) to wire which is securely fastened to the ceiling. The entire wheel acts as a swinging pendulum. Allow the pendulum to swing. Observe how it behaves. Spin the wheel and hold it in a vertical position, then allow it to swing back and forth. Repeat, allowing the wheel to first spin in a horizontal plane. This could lead into a discussion on why a moving bicycle is easier to balance than one at rest.

    Interestingly enough, the gyroscopic effect of the spinning wheels only contributes slightly to the extra stability of a moving bicycle.