# Homework Help: Pebble falling into water; drag force

1. Jan 1, 2013

### unscientific

1. The problem statement, all variables and given/known data

The drag force experienced by the spherical pebble in water is given by: (0.5)CpwAv2

mass of pebble = 83.8g
radius of pebble = 2 cm
C = 0.7

(a) Write down the differential equation governing pebble's descent. (done)

(b) Derive an expression for terminal velocity and evaluate it. (done)

(c) The pebble is dropped 5m above surface of lake, which is also 5m deep. Show that the pebble travels about 10cm in water before reaching it's terminal velocity.

3. The attempt at a solution

Part (a)

Part (b)

Part (c)

2. Jan 1, 2013

### Staff: Mentor

In your exact analysis, the ball will never reach its terminal velocity - but it will approach it with a typical timescale, and you can determine this and show what it does not travel more than 10cm before it reaches a velocity very close to the terminal velocity.

3. Jan 1, 2013

### unscientific

Is there anything wrong with my working in part (b)?? I substituted for v2 using the differential equation..

Last edited: Jan 1, 2013
4. Jan 1, 2013

### Staff: Mentor

I don't see any result in c, and I don't understand what you are doing there.

5. Jan 1, 2013

### unscientific

sorry i meant part (b)!

6. Jan 1, 2013

### SteamKing

Staff Emeritus
1. Are you sure that C is the drag coefficient which uses the entire surface area of the pebble or just the cross-sectional area of the pebble?

2. You recognize that the acceleration of the pebble is x double dot. What about the velocity of the pebble? Can't the velocity also be represented in terms of x?

7. Jan 1, 2013

8. Jan 1, 2013

### haruspex

Sorry, misunderstood the point you were making (and for some reason it's not letting me edit the post).

9. Jan 1, 2013

### Staff: Mentor

You need to substitute x double dot = dv/dt, and solve for the velocity vs time first. Also, what happened to the buoyant force?

10. Jan 2, 2013

### SteamKing

Staff Emeritus
The point about drag coefficients is that they are usually based on the cross sectional or frontal area of the body, not the total surface area.

The OP has used (4/3)pi*r^2, which is the total area of the pebble.

11. Jan 2, 2013

### haruspex

That's not how I read it. Seems to have written the mass as (4/3)pi*r^3*density, then cancelled one r and π with the expression for cross-sectional area. Not that there was much point in doing that: the mass is given, the density is not.

12. Jan 2, 2013

### unscientific

sorry, density of pebble is 2.5g/cm3!

13. Jan 3, 2013

### unscientific

I'm more concerned about the integration part; to find the work done against resistance. Is there anything wrong with that?

14. Jan 3, 2013

### haruspex

When you lost k from your equations, you might have realised you were off course. K must feature in the answer. There are errors further along, but they're not relevant to finding a correct solution.
i don't think you can hope to integrate v2dx straight off. Work with the forces and get the differential equation for the acceleration.

15. Jan 5, 2013

### unscientific

There are 2 ways to approach this problem:

Method 1

Substitute v2 using the differential equation and integrate; which was what I did but I'm not sure what's wrong with my working...

Method 2

a) Solve the differential equation to obtain v in terms of x. show that when x = 0.1 m, v = vterm

16. Jan 5, 2013

### haruspex

As I said, that substitution lost k, and as soon as that happened you were going nowhere. At best you'd end up with tautology. As it happens, you must have made some other error, but I don't think it's interesting to figure out what.

You have $\ddot{x}=\dot{v}=g-kv^2$. You could solve that to find v in terms of t, but there is a trick to eliminate t first and solve for v in terms of x. Is that what you mean?

17. Jan 5, 2013

### Staff: Mentor

Guys,

You've left out the buoyant force on the pebble.

Chet

18. Jan 5, 2013

### haruspex

Good catch!

19. Jan 6, 2013

### unscientific

yeah. replace dv/dt by v(dv/dx) then solve the diferential equation, putting initial condition as v = √2gh.

then substitute v = vterm to find that x = 0.1 m...

20. Jan 6, 2013

### unscientific

Ignore the buoyant force on the pebble.