# The Song Remains the Same

1. Sep 29, 2004

### physicsss

Help!!!

A motorboat traveling at a speed of 2.4 m/s shuts off its engines at t = 0. How far does it travel before coming to rest if it is noted that after 3.0 s its speed has dropped to half its original value? Assume that the drag force of the water is proportional to v.

Last edited: Sep 29, 2004
2. Oct 1, 2004

....anyone?

3. Oct 1, 2004

### Tide

The equation of motion would be
$$\frac {dv}{dt} = -k v$$
Can you integrate that?

4. Oct 1, 2004

### physicsss

Do I get lnv=-kt ? What do I need to do after I find what k is?

Last edited: Oct 1, 2004
5. Oct 1, 2004

-Cheers.

6. Oct 1, 2004

### physicsss

The answer is 10m, and i have no idea how they got it

7. Oct 1, 2004

### poolwin2001

When you solve the differential eqn you will have 2 variables(1 from integration & other 'k')Use the initial conditions given to find them.
At t=0,v=? and one more.
Else,if you did definite integration ,you have to figure out k by the 2nd condition given.

8. Oct 1, 2004

### Tide

$$v = v_0 e^{-kt}$$

9. Oct 1, 2004

### HallsofIvy

Staff Emeritus
Since ln v=-kt+ C (you forgot to add the constant), v= Ce-kt which has two unknown parameters, C and k. Now use the information you were given: "traveling at a speed of 2.4 m/s shuts off its engines at t = 0". Okay, when t=0, v= Ce-k(0)= C= 2.4. "it is noted that after 3.0 s its speed has dropped to half its original value" Okay, when t= 0, v= "half its original value" which is 2.4/2= 1.2 m/s. v= 2.4e-k(3)= 1.2 . Solve that for k.