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Ch 05: Applying Newton's Laws

Chapter 5, Problem 2

A small block has constant acceleration as it slides down a frictionless incline. The block is released from rest at the top of the incline, and its speed after it has traveled 6.80 m to the bottom of the incline is 3.80 m/s. What is the speed of the block when it is 3.40 m from the top of the incline?

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Welcome back everybody. We are making observations about a crate that is on the top of a ramp. So let me go ahead and draw out this little diagram for us. I'm gonna put the crate right up here and we are told that when the crew is released it is going to slide down this ramp now, just for convention sake here, actually going to define my axes here, so I'm going to have this direction B are positive X axis and this direction B are positive Y axis. Now we're told that the length of the ramp is 3.4 m long. We're told that the crate starts out at rest so its initial velocity will be zero and that it hits the bottom of the crate at a final velocity of 2. m/s. And we are tasked with finding what the velocity is at a point half way down the ramp. Now half of 3.4 is 1.7. So this is the velocity that we are trying to find on top of all of this. We are told that the box moves with a constant acceleration but we don't exactly know what that acceleration is. So first and foremost, before figuring out what our velocity, Sorry our velocity is going to be. We should first determine what our acceleration is going to be. So let's use a kid a magic formula for this hour. Kinnah Matic formula states that our final velocity squared is equal to our initial velocity Plus two times our acceleration times the distance that is traveled. In this case it will be a final x minus our initial X A k. It's just gonna be this value right here. Now what we can do is we can use the values already given to find our acceleration and then we can use this formula again with our new distance traveled to figure out our new velocity. So let's first start out with finding our acceleration. So in order to manipulate this formula, I'm going to subtract sorry, V not from both sides of this thing and my apologies. This is actually V not squared. V naught squared. That is important for our formula And then I will divide both sides. These terms are gonna cancel out by two times our distance traveled, two times our distance traveled. This gives us that our acceleration is equal to our final velocity squared minus our initial velocity squared. All divided by two times our distance traveled. Let's go ahead and plug in our values. Final velocity is going to be 2.25 squared minus our initial velocity of zero. All divided by two times our distance traveled of 3.4. This gives us an acceleration of 0.74 m per second squared. Right now that we have found that let's use this initial formula again, I'll rewrite it out. Don't you worry in order to find our velocity when the distance travel is now 1.7. So remember our final velocity or our velocity in this case at 1.7 squared is equal to our initial velocity plus two times our acceleration minus this distance squared. So I'm actually gonna take the square root of both sides and you'll see on the left hand side here that gets rid of our radical and then we are just left with our desired velocity. So let's go ahead and plug in some values, shall we? We have the square root of our initial velocity is still zero. So this will be zero plus two times our acceleration of 0.74 times our new distance traveled of 1.7 mi meters. Because remember we're trying to find the velocity halfway down the ramp when we plug this into our calculator, we get that our velocity at a distance of 1.7 m down the ramp is 1.6 m per second, which corresponds to our answer choice of the Thank you all so much for watching. Hope this video helped. We will see you all in the next one