Answer to Question #90323 in Classical Mechanics for Mathew Hany

Question

I was given this problem: "A 50kg object was held 10m above water, when it was dropped, find the velocity when it reached the surface of the water, and if the object stopped 2.45m under the surface of the water, find the water resistance force."

I was able to solve the first part and find the velocity, which was 14 m/s, but I'm struggling to solve the second part.

The answer my book gave was -2000N for the water resistance, and I don't understand how that is true.

My understanding is that using the equations of motion I could find the acceleration of the object under water, which is -40 m/s², and this means that the net force that is affecting the object is -2000N, and the weight is 490N, so now the resistance must be -2490N? But the book says it's -2000N and I don't know how!

Answer 1

Given:

$$m = 50 kg$$

$$h = 10m$$

$$d = 2.45 m$$

Solution:Part 1

Initial energy of the object is:

$$E_0 =mgh$$

Energy at the surface:

$$E_s = \frac{mv^2}{2}$$

Ignoring air friction:

$$E_0 = E_s$$

So,

$$mgh = \frac {mv^2}{2}$$

thus:

$$v = \sqrt{2gh}$$

Calculating:

$$v = \sqrt{2gh} \approx \sqrt{2* 9.8 * 10} = 14 m/s$$

Answer: 14m/s

Part 2:

When the object stops under water:

$$v= 0$$ $$a = 0$$

So,

$$F_{net} = 0$$

In this case there are 2 forces acting on the object: gravity and buoyancy.

So,

$$F_{net\ stop} = F_G + F_B = 0$$

During motion there is also water resistance:

$$F_{net\ motion} = F_G + F_B + F_r = F_{net\ stop} + F_r = 0 + F_r = F_r$$

Second part can be solved by calculating work done by resistance force:

$$W = F_r* d$$

$$W= \Delta E = 0 - E_s = -mgh$$

We shouldn't account for potential energy difference during submersion since in our case gravitational force is canceled by bouncy.

So,

$$F_r =- \frac{mgh}{d} \approx -\frac{50 * 9.8 * 10}{2.45} = -2000 N$$

Explanation:

You have correctly found acceleration and total force acting on the object. But you also need to account for buoyancy. Since we don't have volume or density of the object, we should assume that in this problem 'object stopped' means zero velocity and acceleration. In this case weight and buoyancy cancel out and resistance force in 2000 N.

Answer: -2000 N

Reference: https://en.wikipedia.org/wiki/Buoyancy