Question

M d v/d t=m g-k v

is a DE that describes the velocity v of a falling mass subject to air resistance proportional to instantaneous velocity where where k>0 is a constant of proportionality. The positive direction is downward. (a) Solve the equation subject to the initial condition v(0)=v₀

Answer 1

To solve the given differential equation m*dv/dt = mg - kv, we can separate the variables and integrate both sides. After solving for the constant of integration, we obtain the final solution mg - kv = (mg - kv₀)e^(-kt).

Step-by-step explanation:

The given equation is a differential equation that describes the velocity of a falling mass subject to air resistance. To solve the equation, we can separate the variables. Rearranging the equation, we have:

m dv/dt = mg - kv

Dividing both sides by (mg-kv), we get:

(1/(mg-kv)) dv = dt

Now, integrate both sides:

∫(1/(mg-kv)) dv = ∫dt

Simplifying the integration, we have:

-1/k ln|mg-kv| = t + C

Multiplying both sides by -k and rearranging, we get:

ln|mg-kv| = -kt - C

Exponentiating both sides, we have:

mg-kv = e^(-kt-C)

Simplifying the equation, we get:

mg-kv = Ce^(-kt)

Since v(0) = v₀, we can substitute these values into the equation:

mg - kv₀ = Ce^(-k*0)

mg - kv₀ = C

Substituting C back into the equation, we get the final solution:

mg - kv = (mg - kv₀)e^(-kt)