(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

So the question I've been asked is, that there is a moving car, at time t = 0, moving with a speed of 50 km/h. At t = 0, the car stops accelerating, and thereby slowing down due to two forces:

The air resistance drag force

A friction which is created by some magnets in the motor (description of this in section 2).

Write a differential equation for the system, as a function of the cars decceleration.

2. Relevant equations

The air resistance is given by: Fd = -(1/2)pCdAv^{2}. Where p, Cd and A are constants

The friction in this motor is given by: Fe = -k*v. Where k is a konstant. This is basicly just a description of a spring using Hooke's law.

3. The attempt at a solution

Writing a differential equation for this systems seems fairly easy. All the forces that acts on the car is equal to Newton's 2nd law F = ma. The constants in the air resistance is given the letter C = -(1/2)pCdA.

ma = C*v^{2}-k*v

Writing this as a differential equation gives:

m (dv/dt) = C v^{2}- k v

However solving it seems much more difficult. This is what I've come up with so far, but I simply just can't get on:

k v + m (dv/dt) = C v^{2}(+ k v on both sides)

(k v)/(c v^{2})+(m/(c v^{2})) dv = dt (divided by c v^{2}on both sides, and multiplied by dt).

(k/c) (1/v) + (m/c) (1/v^{2}) dv = dt (Rewriting the fractions).

I the integrate both sides, where k/c and m/c are both constants, so I only focus on 1/v and 1/v^{2}, which gives:

(k/c) ln|v| + (m/c) ln|v^{2}| = t + K (Where k is the constant.)

Then I move both of the constants to they other side.

ln|v| + ln|v^{2}| = (t + k) (c/k) (c/m)

then I take the exponential of both sides to get v.

v + v^{2}= e^{(t+c)*(c2/(k m))}

However this just seems completely wrong, because solving the remaining eqution for v just wont make sense. There must be an easier way to do this, but I can't figure it out.

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# Homework Help: Differential Equation of a moving car with drag

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