Thread: 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.

Thanks in advance!

Q: A particle of mass m kg moves in a horizontal straight line from the origin O with initial velocity U i , where i is the unit vector in the direction of motion. A resistive force i acts on the particle, where is a constant and i is the velocity of the particle at time seconds measured from the start of the motion.

(i) Show that the velocity of the particle satisfies the differential equation
,
where is the distance of the particle from .
Hence show that .

(ii) Using (i) or otherwise, show that
.

(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)





as required

then


but at

so

and









as required

(ii)




but at

so

and

so as required.

(iii) Help please!

You've done the hard part!

From , v will be U/2 when . Solve that for x.

Put that value of x into and solve for t.

Thankyou Halls of Ivy, I got