How Do You Calculate Forces and Motion in Fluids and Free Fall?

[muna580]

Okay, I got more problems that I need help with and I don't understand how to do this.

1. A small, spherical bead of mass 2:5 g is released from rest at t = 0 in a bottle of liquid shampoo. The differential equation of motion is
dv/dt = g - (b/m)v

The terminal speed is observed to be 2.6 cm/s.The acceleration of gravity is 9:8 m/s2. Find the value of the constant b in the integral form of the above equation
v = (mg/b) ( 1 - e^(-bt/m))
Answer in units of Ns/m.

2. Find the time it takes to reach 0.53vt. Answer in units of s.

3. Find the value of the resistive force when the bead reaches terminal speed. Answer in units of N.

#1-3 are all related

4. A high diver of mass 61.8 kg jumps of a board 10.4 m above the water.
The acceleration of gravity is 9.8 m/s2. If his downward motion is stopped 4.22 s after he enters the water, what average upward force did the water exert on him? Answer in units of N.

 

[anathema]

Hello! I'm happy to help you with these problems. Let's break them down one by one:

1. To find the value of the constant b, we can use the given information about the terminal speed. We know that when the bead reaches terminal speed, the acceleration will be 0, so we can set dv/dt = 0 in the differential equation. This gives us: 0 = g - (b/m) * 2.6 cm/s. We can convert the acceleration of gravity to cm/s^2 by multiplying by 100, and the mass to kg by dividing by 1000. This gives us: 0 = (9800 cm/s^2) - (b/0.0025 kg) * 2.6 cm/s. Solving for b, we get b = (9800 cm/s^2) / (2.6 cm/s) * 0.0025 kg = 9.5 Ns/m.

2. To find the time it takes to reach 0.53vt, we can use the integral form of the equation: v = (mg/b) * (1 - e^(-bt/m)). We know that at t = 0, v = 0, so we can set this equal to 0.53vt and solve for t. This gives us: 0.53vt = (mg/b) * (1 - e^(-bt/m)). Rearranging, we get: t = (m/b) * (1 - e^(-bt/m)) / (0.53v). Plugging in the values we know, we get t = (0.0025 kg) / (9.5 Ns/m) * (1 - e^(-9.5 * t / 0.0025)) / (0.53 * 0.026 m/s). We can solve this equation numerically to find t, which comes out to be approximately 0.25 seconds.

3. To find the value of the resistive force when the bead reaches terminal speed, we can use the same equation as before, but set v = 2.6 cm/s. This gives us: 2.6 cm/s = (mg/b) * (1 - e^(-bt/m)). Solving for b, we get b = (mg) / (2.6 cm/s * (1 - e^(-

 

[kyle8921]

I would recommend breaking down each problem into smaller, more manageable steps. For the first problem, start by identifying the known values: mass of the bead (2.5 g), acceleration of gravity (9.8 m/s2), and terminal speed (2.6 cm/s). Then, use the given differential equation to solve for the value of b in the integral form. This can be done by plugging in the known values and solving for b. The resulting value will be in units of Ns/m.

For the second problem, use the equation v = (mg/b) ( 1 - e^(-bt/m)) to solve for t when v = 0.53vt. This will give you the time it takes to reach 0.53vt in units of seconds.

In order to find the resistive force, use the equation F = bv, where v is the terminal speed (2.6 cm/s) and b is the value calculated in the first problem. This will give you the value of the resistive force in units of Newtons.

For the fourth problem, start by identifying the known values: mass of the diver (61.8 kg), acceleration of gravity (9.8 m/s2), and time (4.22 s). Then, use the equation F = mΔv/Δt to solve for the average upward force exerted by the water on the diver. This will give you the answer in units of Newtons.

Remember to always double check your units and equations to ensure a correct solution. If you are still having trouble, I would recommend seeking assistance from a physics tutor or your instructor.