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In a quarter-mile drag race, two cars start simultaneously from rest, and each accelerates at a constant rate until it either reaches its maximum speed or crosses the finish line. Car A has an acceleration of11.0 $\mathrm{m} / \mathrm{s}^{2}$ and a maximum speed of 106 $\mathrm{m} / \mathrm{s}$ . Car $\mathrm{B}$ has an acceleration of 11.6 $\mathrm{m} / \mathrm{s}^{2}$ and a maximum speed of 92.4 $\mathrm{m} / \mathrm{s}$ . Which car wins the race, and by how many seconds?

Car B will won the race by $\Delta t=0.216 \mathrm{s}$

Physics 101 Mechanics

Chapter 2

Kinematics in One Dimension

Motion Along a Straight Line

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

University of Sheffield

University of Winnipeg

Lectures

04:34

In physics, kinematics is the description of the motion of objects and systems in the frame of reference defined by the observer. An observer has to be specified, otherwise the term is meaningless.

07:57

In mathematics, a position is a point in space. The concept is abstracted from physical space, in which a position is a location given by the coordinates of a point. In physics, the term is used to describe a family of quantities which describe the configuration of a physical system in a given state. The term is also used to describe the set of possible configurations of a system.

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this problem. We're having a race between to race cars. The distance travelled is gonna be 1/4 mile. And before we do anything else, let's convert that distance two meters. So there's 16.9 r 16 09 meters in a mile, so that gives us a distance of about 403 meters. We have two cars. Car A. They both have an initial velocity. Zero. Bebe not is equal to zero, but they have different accelerations. Car has an acceleration of 11 meters per second squared. Carby has an acceleration 11.6 meters per second squared. They also have different maximum velocities. The maximum velocity for a is 106 meters per second, and the maximum velocity for Karbi is 92.4 meters percent. So we need to figure out who wins the race. So we plug in our distance and solve for T for both cars. We could start with that. We'll see if it works with the maximum velocities, but we know that the 403 meters is equal to 1/2 80 square, the initial velocity being zero. I don't have to plug in that first term. So 1/2 times the acceleration times t squared. If I multiply by two and divide by 11 and take the square root. My time is 8.56 seconds, but we have deceived. Is the CarMax out in that time? Let's calculate the final velocity. The final velocity is just gonna be the initial velocity plus acceleration times time, the initial velocity zero. So my accelerations still is 11 times the time that we just calculated, and we get a maximum speed of 94.2. So that's okay that it has not yet reached its maximum velocity of 106. So this car car A will reach the end of the race in 8.56 seconds. Now let's look at Karbi. Carby has an initial velocity, also of zero. The acceleration is Nick is 11.6, so higher, but it has a slot, a smaller maximum velocity. So first, let's figure out what, um, the time would be if it accelerated the whole time. So 1/2 11.6 meters per second squared times t squared. I rearranged that equation and self for T taking the square root. I get 8.34 seconds. Okay, so if 8.34 seconds actually don't want a circle that yet because I don't know if it's going to work because the acceleration might max out the velocity. So let's figure out what the velocity is at 8.34 seconds and it would be be not plus 80. So v, not a zero acceleration is 11.6 and the time is 8.34 seconds. So we had a velocity of 96.7 meters per second, which is a problem because that is, that exceeds the maximum velocity. So now let's go back to the drawing board and say, Well, how long would it take to get to the maximum velocity? If I dio V is equal to be not plus 80 and plug in the maximum velocity 92.4 is equal to zero plus 11.6 square by the way, 11.6 meters per second squared tens T I did by 92.4 by 11.6 my time until we reach maximum speed is 7.97 seconds. So now how far do we travel in that time? Well, we know the initial velocity is zero. So it's just gonna be 1/2 a T squared. So 1/2 a T squared would be 1/2 11 point 1/2 times 11.6 time, 7.97 seconds squared. That gives us a distance of 368.4. So the remaining distance that we have to travel because the total distance is 403 is gonna be 368.4, minus that speed that distance, that we're gonna be traveling at a constant speed. So called ex constant. So subtracting 403 and 3 68.4 I get 34. The distance travelled at a constant speed would be 34.6 meters. So what would that Constance BB? It would be the maximum which is 92.4. So we know that distance equals rate times time, so time would be distance divided by speed. In this case, 34.6 over 92.4. So the time that the car spends travelling at a constant speed is just 8.34 seconds. Sorry 0.374 seconds. I got ahead of myself. 0.374 seconds. If you divide 34.6 by 92.4, the time is 0.374 seconds. We have to add this time to the time it took to accelerate to a speed of 92.4. So 7.97 seconds plus 0.374 seconds is eight 0.34 seconds. And B Seems like it has a smaller time. This is T total for B, I guess if we go way back to what we said, that too for a was 8.56 is larger than 8.34 So b wins 8.56 minus 8.34 is 0.22 seconds. So be wins by 0.22 seconds.

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