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326 Chapter 8 | Linear Momentum and Collisions
forces.
Figure 8.5 A graph of force versus time with time along the -axis and force along the -axis for an actual force and an equivalent effective force. The areas under the two curves are equal.
Making Connections: Baseball
In most real-life collisions, the forces acting on an object are not constant. For example, when a bat strikes a baseball, the force is very small at the beginning of the collision since only a small portion of the ball is initially in contact with the bat. As the collision continues, the ball deforms so that a greater fraction of the ball is in contact with the bat, resulting in a greater force. As the ball begins to leave the bat, the force drops to zero, much like the force curve in Figure 8.5. Although the changing force is difficult to precisely calculate at each instant, the average force can be estimated very well in most cases.
Suppose that a 150-g baseball experiences an average force of 480 N in a direction opposite the initial 32 m/s speed of the baseball over a time interval of 0.017 s. What is the final velocity of the baseball after the collision?
(8.28)
(8.29) (8.30) (8.31)
Note in the above example that the initial velocity of the baseball prior to the collision is negative, consistent with the assumption we initially made that the force exerted by the bat is positive and in the direction opposite the initial velocity of the baseball. In this case, even though the force acting on the baseball varies with time, the average force is a good approximation of the effective force acting on the ball for the purposes of calculating the impulse and the change in momentum.
Making Connections: Take-Home Investigation—Hand Movement and Impulse
Try catching a ball while “giving” with the ball, pulling your hands toward your body. Then, try catching a ball while keeping your hands still. Hit water in a tub with your full palm. After the water has settled, hit the water again by diving your hand with your fingers first into the water. (Your full palm represents a swimmer doing a belly flop and your diving hand represents a swimmer doing a dive.) Explain what happens in each case and why. Which orientations would you advise people to avoid and why?
Making Connections: Constant Force and Constant Acceleration
The assumption of a constant force in the definition of impulse is analogous to the assumption of a constant acceleration in kinematics. In both cases, nature is adequately described without the use of calculus.
Applying the Science Practices: Verifying the Relationship between Force and Change in Linear Momentum
Design an experiment in order to experimentally verify the relationship between the impulse of a force and change in linear momentum. For simplicity, it would be best to ensure that frictional forces are very small or zero in your experiment so that the effect of friction can be neglected. As you design your experiment, consider the following:
• Would it be easier to analyze a one-dimensional collision or a two-dimensional collision?
• How will you measure the force?
• Should you have two objects in motion or one object bouncing off a rigid surface?
• How will you measure the duration of the collision?
• How will you measure the initial and final velocities of the object(s)?
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