How Deep Must a Steel Ball Sink in Oil to Reach Velocity v?

[danielI]

Hey!

In a test of resistance to motion in an oil bath, a small steel ball of mass m is released from rest at the surface. If the resistance to motion is given by R = kv where k is a constant, derive an expression for the depth h required for the ball to reach a velocity v.

$$mg - kv = ma = mv\frac{dv}{ds}$$

$$\int_0^hds = \int_0^{v_t}\frac{mv}{mg-kv}dv$$

$$h = -\frac mk\int_0^{v_t}\frac{-kv}{mg-kv}dv$$

$$h = -\frac mk\int_0^{v_t}\frac{mg-kv-mg}{mg-kv}dv$$

$$h = -\frac mk\left[v_t - \int_0^{v_t}\frac{mg}{mg-kv}dv\right]$$

$$h = -\frac mk\left[v + \frac{mg}{k}\int_0^{v_t}\frac{-k}{mg-kv}dv\right]$$

$$h = -\frac mk\left[v + \frac{mg}{k}\ln(mg-kv_t)\right]$$

$$h = -\frac {m^2g}{k^2}}\ln(mg-kv_t)-\frac{mv_t}{k}$$

$$h = \frac {m^2g}{k^2}}\ln\left(\frac{1}{mg-kv_t}\right)-\frac{mv_t}{k}$$

But the answer sheet says
$$h = \frac {m^2g}{k^2}}\ln\left(\frac{1}{1-(kv_t)/(mg)}\right)-\frac{mv_t}{k}$$

I've calculated this two more times and got the same result.

Why is this so?

 

[pizzasky]

What happened to your constant of integration?
Once you have substituted the initial conditions and solved for the constant of integration, you should get the correct answer.

 

[kyle8921]

The difference in the answer may be due to a slight variation in the initial conditions or assumptions made in the problem. It is also possible that the answer sheet may have a typo or mistake. In any case, both expressions are valid and represent the depth required for the ball to reach a velocity v in an oil bath with resistance R = kv.