Question 9

You MAY use a calculator to solve this problem.

A body is coasting to a stop and the only force acting on it is a resistence proportional to its speed, according to the equation ; s(0) = 0, where v₀ is the body's initial velocity (in m/s), v_ƒ is its final velocity, m is its mass, k is a constant, and t is time.

(a) If a body with mass m = 50 kg and k = 1.5 kg/sec initially has a velocity of 30 m/s, how long, to the nearest second, will it take to slow to 1 m/s?

(b) How far, to the 10 nearest meters, will the body coast during the time it takes to slow from 30 m/s to 1 m/s?

(c) If the body coasts from 30 m/s to a stop, how far will it coast?

Solution

 (a) The final speed is 1m/s and the original or initial speed is 30m/s. So we just plug these in the derivitive equation and solve for time (t).

 Therefore it takes about 113 s to slow down to 1m/s. We can also check this answer by using the points (113,1).

answers are equal, so th etime it takes to slow down (113s) is correct.

(b) First off we have to find the parent function. We do this by seperating the variables, antidifferentiating, and than solving for the s(t).

This equation looks odd with the weird exponent, but i don't know what I did wrong, if I even did anything wrong. It doesn't look right.

Well now we have to solve for C. For this we need another point on the parent function like (0,0), but I can't seem to solve for another point. I'm not sure if we can use the speed formula (v=d/t), which would give us the point (113,113). However that does not seem correct.

Anyways once we find that point we than solve for the constant k by plugging in all the numbers into the parent equation. Now we can solve for s(113) since it takes 113s to slow from 30m/s to 1m/s.

(c) First we have to find the time it takes to slow down all the way to 0m/s from 30m/s. Then we can use that time and just plug it into the parent function, and we have our answer.

will be solved by crystal =)