The kinetic energy of a rolling billiard ball is given by KE = 1/2 mv². Suppose a 0.17-kg billiard ball is rolling down a pool table with an initial speed of 4.5 m/s. As it travels, it loses some of its energy as heat. The ball slows down to 3.8 m/s and then collides head-on with a second billiard ball of equal mass. The first billiard ball completely stops and the second one rolls away with a velocity of 3.8 m/s. Assume the first billiard ball is the system. Calculate q.

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Welcome back everyone. Kinetic energy of objects can be calculated using the formula K equals one half mv squared of 5.5 kg cart is traveling with an initial speed of 15.7 m per second. And after a while it slows down to a speed of 10.9 m per second and it loses some of its kinetic energy as heat. We need to calculate the amount of heat lost by the cart. Let's begin by recalling our variable for heat which is represented as Q. And because the prompt tells us that our kinetic energy of the cart is released as heat or lost as heat, we can say that Q is equal to our change in kinetic energy represented by delta K. E. And what this is interpreted as is our kinetic energy of the cart once it slows down. So at its final position minus the kinetic energy of the cart when it takes off initially. And so because the prompt tells us that kinetic energy is calculated as one half M. V squared. We can reinterpret this so that we can say Q is equal to 1/2 and we'll use colors. So we have one half times the mass of the cart, times its velocity squared for the final conditions. Once it's slowing down subtracted from one half times the mass of the car, times its velocity squared before the car takes off. So in its initial conditions. And so we can further simplify this because we, according to the prompts will have the same mass of the cart in its final and initial state. So the only thing changing in our formula is velocity and so we can simplify this to say that Q is equal to I'm sorry, this is an equal sign. Q is equal to 1/2 Times the mass given as 5.5 kg. And then we would just take the difference in the velocities. So we would say v squared final -3 Squared initial. And so plugging in our velocity we would say Q is equal to We can simplify this first step here. So this product will give us a value of 2.75 kg which is then multiplied by our difference in our velocities. So for the final velocity of the cart, once it's slowing down we're given according to the prompted value of 10.9 m/s which is squared. And then we would subtract this from the initial conditions of the cart. So according to the prompt, The cart has an initial speed of 15.7 m/s and we want to square this. So this should be in brackets because we're going to focus on doing this first. So in our next line will say that the Heat lost Q is equal to we have 2.75 kg Which is going to be multiplied by the difference where the result of the difference which is a value of negative 127.68 and we have units of meters squared divided by seconds squared. And so now we just want to take the product of these two terms. And what we'll find is that we have a value for heat equal to negative 351.12. And our units are kilograms times meters squared divided by seconds squared. And we should recall that one jewel is actually equivalent to one kg times meter squared times seconds squared. So these units are interchangeable. And we can ultimately say that heat is equal to negative 351. jewels. But going back to our prompt, we want to make sure we have the correct amount of minimum sick fix and we can see we have a minimum of two sig figs given by the mass of the cart 5.5 kg. And so we would round this to about 26 fix and that would give us a value of 3. which is negative times 10 to the second power jewels. And this would be our final answer for the amount of heat lost by the cart, which will correspond to choice D in the multiple choice. I hope that everything I reviewed was clear. If you have any questions, please leave them down below and I'll see everyone in the next practice video

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Key Concepts

Kinetic Energy

Conservation of Momentum

Heat Transfer (q)