what is solution to mv' = -9.8m + kv with respect to t ? v(0)=v(0)?
a. v(t) = 9.8m/k + (v(0)+9.8m/k)e^(kt/m)
b. a. v(t) = 9.8m/k + (v(0) - 9.8m/k)e^(kt/m)
- Anonymous1 decade agoFavorite Answer
mv' - kv = - 9,8
This is a non homogen differential equation
You have to solve first the homogen differential equation:
mv' - kv = 0
mv' = kv
m dv/dt = k v
m dv/v = k/dt
int m dv/v = int k/dt
m ln v = kt
ln v = kt/m + C
v = e^kt/m + e^C = A e^kt/m
Now you have to calculate a particular solution from the non homogen differential equation and add both solutions.
mv' = - 9,8 + kv accepts v = B,
B a constant, as a particular solution
0 = - 9.8 + k. B
So, B = 9.8/k
Now add both solutions:
v = A e^kt/m + 9.8/k
And use the initial conditions: v(0) = v(0) to evaluate A
v(0) = A e^0 + 9.8/k
So, A = v(0) - 9.8/k
The solution is b. You mistyped it too.
- 1 decade ago
1) Rewrite in terms of dv/dt.
m(dv/dt) = -9.8m + kv
2) Rearrange so v and dv are on the same side. This is a separable differential equation.
[1/(kv - 9.8m)]dv = [1/m]dt
3) Integrate both sides
[ln(kv - 9.8m)]/k = (t/m) + C
4) ln(kv - 9.8m) = (tk/m) + C
5) kv - 9.8m = e^ [(tk/m) + C]
6) kv = Ce^ [tk/m] + 9.8m
7) v = [Ce^ [tk/m] + 9.8m]/k
I don't think this matches A or B, so C.
- 1 decade ago
take S as the symbol of integra
S m(dv)=S -9.8m(dt)+S kv(dt)
v is a function of t=> v(t)=(-9.8mt)/(m-kt)
I think the answer is "neither"
- Anonymous1 decade ago
Please put this in standard notaion....NO one, including myself, knows what the HECK you are talking about..