# Thread: Forces and calculus problem

1. ## Forces and calculus problem

Here is a problem i've been working on. I think i've got the first two parts but I'm stuck on the last section. I'd appreciate anyone checking my solution so far and even better helping me with the third section.

Q: A particle of mass m kg moves in a horizontal straight line from the origin O with initial velocity U i $\displaystyle ms^{-1}$, where i is the unit vector in the direction of motion. A resistive force $\displaystyle -mkv^3$i acts on the particle, where $\displaystyle k$ is a constant and $\displaystyle v$i is the velocity of the particle at time $\displaystyle t$ seconds measured from the start of the motion.

(i) Show that the velocity of the particle satisfies the differential equation
$\displaystyle \frac{dv}{dx} = -kv^2$,
where $\displaystyle x$ is the distance of the particle from $\displaystyle O$.
Hence show that $\displaystyle v = \frac{U}{1 + kUx}$.

(ii) Using (i) or otherwise, show that
$\displaystyle kUx^2 + 2x = 2Ut$.

(iii) Find an expression, in terms of k and U, for the time taken for the speed of the particle to reduce to half its initial value.

Solution:

(i) $\displaystyle F = -mkv^3$

$\displaystyle ma = -mkv^3$

$\displaystyle v\frac{dv}{dx} = -kv^3$

$\displaystyle \frac{dv}{dx} = -kv^2$ as required

then
$\displaystyle \int \frac{1}{kv^2} dv = \int - dx$

$\displaystyle - \frac{1}{kv} = -x + c$ but at $\displaystyle x=0 , v=U$

so $\displaystyle c = -\frac{1}{kU}$

and $\displaystyle \frac{1}{kv} = -x - \frac{1}{kU}$

$\displaystyle -1 = -kvx - \frac{kv}{kU}$

$\displaystyle kvx - 1 = -\frac{v}{U}$

$\displaystyle kvxU + v - U = 0$

$\displaystyle v(kUx + 1) = U$

$\displaystyle v = \frac{U}{1+kUx}$as required

(ii)
$\displaystyle \frac{dx}{dt} = \frac{U}{1+kUx}$

$\displaystyle \int (1 + kUx)dx = \int U dt$

$\displaystyle x + \frac{1}{2} kUx^2 = Ut + c$ but at $\displaystyle x=0, t=0$

so $\displaystyle c=0$

and $\displaystyle \frac{1}{2} kUx^2 + x = Ut$

so $\displaystyle kUx^2 + 2x = 2Ut$ as required.

From $\displaystyle v= \frac{U}{1+ kUx}$, v will be U/2 when $\displaystyle \frac{U}{2}= \frac{U}{1+ kUx}$. Solve that for x.
Put that value of x into $\displaystyle kUx^2+ 2x= 2Ut$ and solve for t.
3. Thankyou Halls of Ivy, I got $\displaystyle t=\frac{4}{kU^2}$