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Question
a car moving with speed v on a straight road can be stopped with in distance d on applying brakes. if the same car is moving with speed 3v and brakes provide half retardation , then car will stop after travelling distance [ option:1)6 d  2)3 d  3) 9 d  4)18 d ]    sir, i tired a lot and i got  3d but the correct option is 18 d

Answer
Hello saroj,

You need to use the principle of conservation of energy. When moving at speed v, the kinetic energy of the car is
KE = (1/2)*m*v^2
The work the car's brake system does while stopping the car is the braking force, F, times the distance, d, the car travels while stopping.
W = F*d
To stop the car, the brakes have to do an amount of work equal to the car's kinetic energy. So
KE = W
(1/2)*m*v^2 = F*d
So you can write an expression for d. I'll leave that to you: solve for distance d.

In the second case, when moving at speed 3*v, the kinetic energy of the car is
KE = (1/2)*m*(3*v)^2 = (1/2)*m*9*v^2
The braking system's braking force is now F/2. Therefore work the car's brake system does while stopping the car is the braking force, F/2, times the distance, d', the car travels while stopping in this case.
W = (F/2)*d
To stop the car, the brakes have to do an amount of work equal to the car's kinetic energy. So
KE = W
(1/2)*m*9*v^2 = F*d'/2
Now solve for new distance d' and see by what factor this distance is different.

I hope this helps,
Steve

Edit: OK, trouble with the math. I can do the math.
1st case:
(1/2)*m*v^2 = F*d ...... Solve for distance d.

d = (1/F)*(1/2)*m*v^2  ..........  Eq 1

2nd case:
(1/2)*m*9*v^2 = F*d'/2
Now solve for new distance d'.
d' = 2*9*(1/F)*(1/2)*m*v^2 = 18*(1/F)*(1/2)*m*v^2 .......... Eq 2

I've grouped the string of things in Equations 1 and 2 above to make this next step easier. Notice in Equation 2 that everything from (1/F) to the end is bit for bit what Equation 2 says d is equal to.
So, you can substitute d for (1/F)*(1/2)*m*v^2. And that gives you
d' = 18*d

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Steve Johnson

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I would be delighted to help with questions up through the first year of college Physics. Particularly Electricity, Electronics and Newtonian Mechanics (motion, acceleration etc.). I decline questions on relativity and Atomic Physics. I also could discuss the Space Shuttle and space flight in general.

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I have a BS in Physics and an MS in Electrical Engineering. I am retired now. My professional career was in Electrical Engineering with considerable time spent working with accelerometers, gyroscopes and flight dynamics (Physics related topics) while working on the Space Shuttle. I gave formal classroom lessons to technical co-workers periodically over a several year period.

Education/Credentials
BS Physics, North Dakota State University
MS Electrical Engineering, North Dakota State University

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