In this lab we explore the forces of interaction between two objects and study the changes in motion that result from these interactions. We are especially interested in studying collisions and explosions in which interactions take place in fractions of a second. Early investigators spent a considerable amount of time trying to observe collisions and explosions, but they encountered difficulties. This is not surprising, since the observation of the details of such phenomena requires the use of instruments—such as high-speed cameras—that were not yet invented.
However, the principles describing the outcomes of collisions were well understood by the late seventeenth century when several leading European scientists, including Isaac Newton, developed the concept of quantity-of-motion to describe both elastic collisions in which objects bounce off each other and inelastic collisions in which objects stick together. These days we use the word momentum rather than quantity-of-motion to help us understand the nature of collisions and explosions.
We will begin our study of collisions by exploring the relationship between the forces experienced by an object and its momentum change. It can be shown mathematically from Newton’s laws and experimentally from our own observations that the change in momentum of an object is equal to a quantity called impulse. Impulse takes into account both the magnitude of the applied force at each instant in time and the time interval over which this force acts. The statement of equality between impulse and momentum change is known as the impulse–momentum theorem.
Originally, Newton did not use the concept of acceleration or velocity in his laws. Instead, he used the term “motion,” which he defined as the product of mass and velocity (the quantity we now call momentum). Let’s examine a translation from Latin of Newton’s first two laws with some parenthetical changes for clarity.
The more familiar contemporary statement of the second law is that the net force on an object can be calculated as the product of its mass and its acceleration where the direction of the force and of the resulting acceleration are the same. Newton’s statement of the second law and the more modern statement are mathematically equivalent.
Now let’s test your intuition about momentum and forces. You are sleeping in your room, and your younger brother is making too much noise outside. To keep the noise out, you want to close the door. The room is so messy that you cannot get to the door. The only way to close the door is to throw either a blob of clay or a bouncy ball at the door.
To complete the next activity you will need:
BE SURE TO PLUG IN THE DONGLE, TURN ON THE IOLAB AND CALIBRATE YOUR FORCE SENSOR NOW, BEFORE MOVING ON TO THE NEXT SLIDE!
First some predictions.
Prediction 1-1: Assuming the clay blob and the bouncy ball have the same mass, and that you throw them with the same velocity, which would you throw to close the door—the clay blob, which will stick to the door, or the bouncy ball, which will bounce back at almost the same speed as it had before it collided with the door?
Prediction 1-2: Give reasons for your choice using any notions you already have or any new concepts developed in physics, such as force, energy, momentum, or Newton’s laws. If you think that there is no difference, justify your answer. Remember, your life depends on it!
You can test your predictions by dropping sticky clay balls and the bouncy ball from equal heights onto the plate attached to the force sensor. Before testing your predictions, you must measure the weight of each object:
Bouncy Ball weight:
Med. Clay ball weight:
Large Clay weight:
Before testing your predictions, you must calculate the mass of each object:
Bouncy Ball mass:
Med. Clay ball mass:
Large Clay mass:
Now you are ready to test your predictions.
Question 1-1: What is the maximum force exerted by the bouncy ball?
Question 1-2: What is the maximum force exerted by the medium clay ball?
Question 1-3A: Did your observations agree with your prediction? Which resulted in a bigger maximum force—the bouncy ball or clay?
Question 1-3B: Based on your observations, which should you throw at the door—the bouncy ball or the clay?
Question 1-3C: Explain based on your observations.
Prediction 1-3: What will happen to the maximum force if you increase the mass of the ball of clay but allow it to collide with the same velocity (drop it from the same height)?
Prediction 1-4: What will happen to the maximum force if the velocity just before impact is increased by dropping the ball from a greater height?
Test your Prediction:
Question 1-4A: Did your observations agree with your predictions? Explain.
Question 1-4B: What factors seem to determine the maximum force exerted on the force sensor?
It would be nice to be able to use Newton’s formulation of the second law of motion to find collision forces, but it is difficult to measure the rate of change of momentum during a rapid collision without special instruments. However, measuring the momenta of objects just before and just after a collision is not usually too difficult. This led scientists in the seventeenth and eighteenth centuries to concentrate on the overall changes in momentum that resulted from collisions. They then tried to relate changes in momentum to the forces experienced by an object during a collision.
In the next activity you are going to explore the mathematics of calculating momentum changes for the two types of collisions—the elastic collision, where the ball bounces off the door, and the inelastic collision, where the clay blob sticks to the door.
Comment: Recall that momentum is defined as the vector quantity ; i.e., it has both magnitude and direction. Mathematically, momentum change is given by the equation where is the initial momentum of the object just before a collision and is its final momentum after the collision. Remember, in one dimension, the direction of a vector is indicated by its sign.
Prediction 1-5A: Which object undergoes the greater momentum change during the collision with a door—the clay blob or the bouncy ball?
Prediction 1-5B: Explain your reasoning carefully.
Check your prediction with some calculations of the momentum changes for both collisions that you carried out. This is a good review of the properties of one-dimensional vectors. Carry out the following calculations for the original height and original mass of both the clay ball and bouncy ball.
Question 1-5: Compare your calculated changes in momentum to your predictions. Do they agree? Which object had the larger change in momentum?
Question 1-6: How does change in momentum seem to be related to the maximum force applied to the ball? Note: Assume that the time intervals the clay ball and bouncy ball are in contact with the IOLab are the same.
You have observed that the ball that bounces off the door will exert a larger maximum force on the door than the clay ball, which sticks to it. The ball that bounces off the door has the larger change in momentum because it reverses direction, while the clay ball merely comes to rest stuck to the door.
However, just looking at the maximum force exerted on the ball does not tell the whole story. You can see this from a simple experiment tossing raw eggs. We will do it as a thought experiment to avoid the mess! Suppose somebody tosses you a raw egg and you catch it. (In physics jargon, one would say in a very official tone of voice, “The egg and the hand have undergone an inelastic collision.”) What is the relationship between the force you have to exert on the egg to stop it, the time it takes you to stop it, and the momentum change that the egg experiences? You ought to have some intuition about this matter. In more ordinary language, would you want to catch the egg slowly (by relaxing your hands and pulling them back) or quickly (by holding your hands rigidly)?
Question 1-7A: Suppose the time you take to bring the egg to a stop is Δt. Would you rather catch the egg in such a way that Δt is small or large?
Question 1-7B: Explain why.
Question 1-8: What do you suspect might happen to the average force you exert on the egg while catching it when Δt is small?
In bringing an egg to rest, the change in momentum is the same whether you use a large force during a short time interval or a small force during a long time interval. Of course, which one you choose makes a lot of difference in whether the egg breaks or not!
A quantity called impulse may have been defined for you in lecture and/or in your textbook. It combines the applied force and the time interval over which it acts. In one dimension, for a constant force acting over a time interval Δt, as shown in the graph below on the left, the impulse is:
As you can see, a large force acting over a short time and a small force acting over a long time can have the same impulse.
Note that FΔt is the area of the rectangle, i.e., the area under the force vs. time curve. If the applied force is not constant, then the impulse can still be calculated as the area under the force vs. time graph, as seen in the graph above on the right. It is the impulse that equals the change in momentum:
Let’s first see qualitatively what an impulse curve might look like in a real collision in which the forces change over time during the collision. To explore this idea you will need:
First some predictions.
Attach the spring and screw to the IOLab. Place the IOLab so that the wheels are on a smooth horizontal surface. Shove the IOLab so it is coasting and allow it to hit the wall. Observe what happens to the spring.
Prediction 2-1: If friction is negligible, what is the net force exerted on the IOLab just before it starts to collide?
Prediction 2-2: When is the net force on the cart maximum?
Prediction 2-3: Roughly how long does the collision process take?
Prediction 2-4: Remembering what you observed, choose the one graph below that best fits the force the wall exerts on the IOLab as a function of time during the collision. The vertical axis is the force in N and the horizontal axis is time in seconds.
During the collision the force is not constant. To measure the impulse and compare it to the change in momentum of the IOLab, you must (1) plot a force–time graph and find the area under it, and (2) measure the velocity of the cart before and after the collision with the wall. Fortunately, the IOLab will allow you to do this. You will need:
In a perfectly elastic collision between a cart and a wall, the cart would recoil with exactly the same magnitude of momentum that it had before the collision. Because your cart’s spring bumper is not perfect, you can only produce a nearly elastic collision.
Weight:
Mass:
Question 2-1: Does the shape of the force–time graph agree with your Prediction 2-1? Explain.
J:
Initial velocity toward the wall:
Final velocity away from the wall:
∆p:
Question 2-2A: Did the calculated magnitude of the change in momentum of the IOLab equal the measured impulse applied to it by the wall during the nearly elastic collision?
Question 2-2B: Explain.
What would the impulse be if the initial momentum of the IOLab were larger? You will find out in the following activity. First some predictions.
Prediction 2-5A: What would the impulse be if the initial momentum of the IOLab were larger?
Prediction 2-5B: Explain.
Equipment you will need:
You can assemble a mass equal to the mass of the IOLab in the following way:
Average force (weight of box with masses):
Test your prediction.
New mass of IOLab:
Velocity toward the wall:
Velocity away from the wall:
Calculate the initial and final momenta. show your work. Be careful of signs--remember that the direction away from the wall is positive.
Calculate the change in momentum. show your work.
Impulse:
Question 2-3: Did the impulse agree with your prediction? Explain.
Question 2-4: Were the impulse and change in momentum equal to each other? Explain why you think the results came out the way they did.
It is also possible to examine the impulse-momentum theorem in a collision where the cart sticks to the wall and comes to rest after the collision. This can be done by replacing the spring with some clay.
Prediction 2-6A: What would the impulse be if the collision were inelastic rather than elastic, i.e, what if the cart stuck to the wall after the collision?
Prediction 2-6B: Explain.
In addition to the equipment you have used so far, you will need:
Now when the cart hits the wall, it will come to rest stuck to the clay. What do you predict about the impulse?
Prediction 2-7A: Will it be the same, larger, or smaller than in the nearly elastic collision?
Prediction 2-7B: What do you predict now about the impulse and the change in momentum? Will they equal each other, or will one be larger than the other?
Prediction 2-7C: Explain your reasoning.
Velocity toward the wall:
Velocity away from the wall:
Calculate the initial and final momenta. show your work. Be careful of signs--remember that the direction away from the wall is positive.
Calculate the change in momentum. show your work.
Find the impulse from the Force graph, as in Activity 2-2. show your calculations.
Impulse:
Question 2-5: Compare the force–time curve for the inelastic collision to that for the nearly elastic collision. How are they similar and how are they different?
Question 2-6: Were the impulse and change in momentum equal to each other for the inelastic collision? Explain why you think the results came out the way they did.
Question 2-7: Do you think that the momentum change is equal to the impulse for all collisions? Justify your answer.
Question 2-8: Now think back at throwing a clay blob or a bouncy ball at the door. Do your answers to Questions 1-3 and 1-4 make sense? Justify your answer.
Please remember to edit the report (insert your name - and if necessary your partners), export the report and submit it on D2L.
Now do the homework associated with this lab.