# Thread: a body is coasting to a stop and the only force acting on it is a resistance

1. ## a body is coasting to a stop and the only force acting on it is a resistance

A body is coasting to a stop and the only force acting on it is a resistance proportional to its speed, according to the equation:
$\displaystyle \frac{ds}{dt}=v_f=v_0e^{(\frac{-kt}{m})}$ ; s(0)=0, where $\displaystyle v_0$ is the body's initial velocity (in m/s), $\displaystyle v_f$ 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 nearest 10 meters, will the body coast during the time 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?

2. You can get (a) by using the given differential equation and solving for t.

Since $\displaystyle v_0\text{, }k,\text{ and }m$ are constants, you have $\displaystyle \frac{ds}{dt}$ equal to a function of t, so you only have to take the integral of both sides to get s as a function of t.

If you don't know how to integrate, try setting $\displaystyle s=c_1e^{c_2t}+C$, take the derivative and choose values of $\displaystyle c_1\text{ and }c_2$ to make the equation work. You'll have to use the initial condition s(0)=0 to get C.

The answer to (c) is that the question is invalid - if you look at the differential equation, you see that $\displaystyle v_f$ never reaches zero, so it never stops. But you can take the limit as t goes to infinity of the equation you found in (b) to get the result.

Post again in this thread if you're still having trouble.

3. Originally Posted by yoman360
A body is coasting to a stop and the only force acting on it is a resistance proportional to its speed, according to the equation:
$\displaystyle \frac{ds}{dt}=v_f=v_0e^{(\frac{-kt}{m})}$ ; s(0)=0, where $\displaystyle v_0$ is the body's initial velocity (in m/s), $\displaystyle v_f$ 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 nearest 10 meters, will the body coast during the time 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?

(a) solve the equation $\displaystyle v(t) = v_0 e^{-\frac{kt}{m}}$ for $\displaystyle t$ , then sub in your given values to calculate $\displaystyle t$

(b) using $\displaystyle t$ found in part (a) as the upper limit of integration ...

$\displaystyle d = \int_0^t v(t) \, dt$

(c) $\displaystyle d = \lim_{b \to \infty} \int_0^b v(t) \, dt$

4. Originally Posted by skeeter
(a) solve the equation $\displaystyle v(t) = v_0 e^{-\frac{kt}{m}}$ for $\displaystyle t$ , then sub in your given values to calculate $\displaystyle t$

(b) using $\displaystyle t$ found in part (a) as the upper limit of integration ...

$\displaystyle d = \int_0^t v(t) \, dt$

(c) $\displaystyle d = \lim_{b \to \infty} \int_0^b v(t) \, dt$
I got t=0 but i think i did something wrong

5. I assume you mean for (a). You should be setting $\displaystyle v_0=30\text{, }k=1.5\text{, }m=50\text{, and }v_f=1$. If you still get t=0, please show us your work and one of us will show you where you went wrong.

- Hollywood

6. Originally Posted by hollywood
I assume you mean for (a). You should be setting $\displaystyle v_0=30\text{, }k=1.5\text{, }m=50\text{, and }v_f=1$. If you still get t=0, please show us your work and one of us will show you where you went wrong.

- Hollywood
$\displaystyle v_f=v_0e^{(\frac{-kt}{m})}$

$\displaystyle 1=30e^{(\frac{-1.5t}{50})}$

when making this post i realized the mistake i made

now i get

$\displaystyle t=\frac{-50}{1.5}*ln(\frac{1}{30})$

(A) t≈ 133 seconds

hopefully thats correct

7. Originally Posted by yoman360
$\displaystyle v_f=v_0e^{(\frac{-kt}{m})}$

$\displaystyle 1=30e^{(\frac{-1.5t}{50})}$

when making this post i realized the mistake i made

now i get

$\displaystyle t=\frac{-50}{1.5}*ln(\frac{1}{30})$

hopefully thats correct
correct ... that is the value of t such that v(t) = 1