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kingjewel1

kinematics

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could you please help me with this question

I would be very much obliged. I cant hack it for my exam

 

A racing car of mass m travels along a straight road. While travelling the racing car is subject to a constant frictional resistance equal to ma (a is a positive constant) and aire resistance equal to k times thee square of its velocity v. The engine of the care can rpovide a constant propelling froce equal to mb (b is a positive constant) and can bring the car to a terminal velocity Vinfinity.

 

a) given that the racing car starts from rest, show that it reaches a speed of Vinfinity/2 in time

[math]\frac{v_{\infty}log3}{2(b-a)}[/math]

 

b) at this point the engine is switched off. Show that the racing car comes to rest in a further time

[math]\frac{v_{\infty}}{\sqrt(a(b-a))}[/math][math]arctan\sqrt\frac{b-a}{4a}[/math]

 

thank you very much in advance

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F = m dv/dt = mb-kv2-ma

 

You want to solve for vinfinity, the case where F = 0, and put that back into the equation.

 

Then, for the second part, solve again setting mb to zero.

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Thank you swansont.

I dont really understand the first part, could you explain it a bit more please.

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Force is mass x acceleration (F=ma), and a = dv/dt, i.e. the time-derivative of velocity. (using "a" as a constant confuses things, so I refrained from using F=ma in the same equation as where the constant appears)

 

So you have a differential equation that needs to be solved, and you set dv/dt equal to the sum of all the forces; since it's a vector, you have to use the proper sign for the terms: + if they cause an acceleration in one direction, - if they cause an acceleration in the other direction. Then math happens.

 

You know that at the terminal velocity the force is zero, which allows you to put the term with k in it in terms of vinfinity, which is what was done in the answer you gave.

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