First, the momentum principle says that a net force changes the momentum of an object where the momentum is the product of mass and velocity. Working in one dimension to avoid dealing with vectors, I can write it like this:
Image may contain Number Text Symbol and Alphabet
If you consult your introductory physics textbook, you'll see that this is essentially the same as Newton's Second Law, which states that the net force is equal to the product of mass and acceleration (where acceleration represents the change in velocity). You can rewrite the momentum principle to solve for the change in momentum (which is useful). It looks like this:
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OK, now for the second big idea, the work energy principle. It states that, for a single particle, the work done on an object is equal to the change in kinetic energy. Work is defined as the product of a force in the direction of a displacement. I can write this as:
Image may contain Text Alphabet Letter Number and Symbol
Just to be clear, Δr represents the displacement (how far the force pushes something) and θ represents the angle between the force and the direction the object moves. As with the momentum principle, I can rewrite this so it looks a bit more useful:
Image may contain Text Word Logo Symbol Trademark and Number
Let's take a second and look at these two ideas. Two things differentiate the momentum principle from the work energy. First, it is technically a vector equation because the momentum of an object depends upon its direction of movement. Second, the momentum principle depends upon the change in time (this is important). The work energy principle depends only on displacement, not time.
Explanation:
Now to my great physics question. Suppose a heavy truck and a light car start with the same momentum (if it makes you happy, we can say the truck has a mass three times that of the car). Both vehicles have the same force acting on them to bring them to a stop. Which one stops first?
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If you want to take a moment to think about this, I'll wait.
I'm still waiting.
OK, hopefully you have an answer by now. If you like, you can check with friends to see what they think. However, since I'm not there and you aren't here, I will just share two common answers people provide.
Answer number 1: The light car stops first. Since it has lower mass, the force acting on it results in larger acceleration. This, in turn, causes the car to slow down more quickly because the truck has a large mass and a small acceleration.
Answer number 2: They stop in the same amount of time. Yes, it's true that the car has a lower mass and a higher acceleration. However, it starts with a much larger velocity since the two vehicles have the same starting momentum. In the end, both vehicles will have the same force with the same change in momentum. According to the momentum principle, they must have the same change in time.
Clearly, answer number 2 is correct. The cars stop at the same time because they start with the same momentum. Just for fun, let's create a numerical calculation for this. Of course, that requires some actual values for the mass of the two vehicles, the starting momentums, and the stopping force. We'll say the car has a mass of 10 kg (it's a really small car) and the truck has a mass of 30 kg (three times the mass of the tiny car). The initial momentum is 20 kg*m/s and the stopping force is 2 newtons.
A plot of the x-velocity for the car and the truck looks like this:
You can see that the car does indeed start with a higher velocity, but both cars stop at the same time. Yes, this is a plot of velocity vs. time instead of distance vs. time for a very particular reason.
Answers & Comments
Answer:
First, the momentum principle says that a net force changes the momentum of an object where the momentum is the product of mass and velocity. Working in one dimension to avoid dealing with vectors, I can write it like this:
Image may contain Number Text Symbol and Alphabet
If you consult your introductory physics textbook, you'll see that this is essentially the same as Newton's Second Law, which states that the net force is equal to the product of mass and acceleration (where acceleration represents the change in velocity). You can rewrite the momentum principle to solve for the change in momentum (which is useful). It looks like this:
Image may contain Logo Symbol Trademark and Word
OK, now for the second big idea, the work energy principle. It states that, for a single particle, the work done on an object is equal to the change in kinetic energy. Work is defined as the product of a force in the direction of a displacement. I can write this as:
Image may contain Text Alphabet Letter Number and Symbol
Just to be clear, Δr represents the displacement (how far the force pushes something) and θ represents the angle between the force and the direction the object moves. As with the momentum principle, I can rewrite this so it looks a bit more useful:
Image may contain Text Word Logo Symbol Trademark and Number
Let's take a second and look at these two ideas. Two things differentiate the momentum principle from the work energy. First, it is technically a vector equation because the momentum of an object depends upon its direction of movement. Second, the momentum principle depends upon the change in time (this is important). The work energy principle depends only on displacement, not time.
Explanation:
Now to my great physics question. Suppose a heavy truck and a light car start with the same momentum (if it makes you happy, we can say the truck has a mass three times that of the car). Both vehicles have the same force acting on them to bring them to a stop. Which one stops first?
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JOHN TIMMER, ARS TECHNICA
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SECURITY
This Ballot-Count Livestream Is the Only Thing Worth Watching
BRIAN BARRETT
If you want to take a moment to think about this, I'll wait.
I'm still waiting.
OK, hopefully you have an answer by now. If you like, you can check with friends to see what they think. However, since I'm not there and you aren't here, I will just share two common answers people provide.
Answer number 1: The light car stops first. Since it has lower mass, the force acting on it results in larger acceleration. This, in turn, causes the car to slow down more quickly because the truck has a large mass and a small acceleration.
Answer number 2: They stop in the same amount of time. Yes, it's true that the car has a lower mass and a higher acceleration. However, it starts with a much larger velocity since the two vehicles have the same starting momentum. In the end, both vehicles will have the same force with the same change in momentum. According to the momentum principle, they must have the same change in time.
Clearly, answer number 2 is correct. The cars stop at the same time because they start with the same momentum. Just for fun, let's create a numerical calculation for this. Of course, that requires some actual values for the mass of the two vehicles, the starting momentums, and the stopping force. We'll say the car has a mass of 10 kg (it's a really small car) and the truck has a mass of 30 kg (three times the mass of the tiny car). The initial momentum is 20 kg*m/s and the stopping force is 2 newtons.
A plot of the x-velocity for the car and the truck looks like this:
You can see that the car does indeed start with a higher velocity, but both cars stop at the same time. Yes, this is a plot of velocity vs. time instead of distance vs. time for a very particular reason.