Terminal Velocity and Bouyancy - F = kv

In summary: So dv/dt = -4.90 N + 1.63s-1 *v. This will give you a differential equation that you can solve to find v(t).In summary, the problem involves a sinking rock in the ocean, acted upon by gravity, buoyancy, and drag forces. The buoyancy is equal to half the weight of the rock, and the drag force is modeled by F = kv. Part a asks for the terminal speed of the rock, which is found to be 3.02m/s. Part b involves calculating the depth, speed, and acceleration of the rock 1.50 seconds after it is released, which requires integrating the net force equation with respect to time. Part c asks for
  • #1
ultimateman
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Homework Statement



A rock of mass 0.400 kg is released from the surface and sinks in the ocean. As the rock descends it is acted upon by three forces: gravity, buoyancy, and drag. The buoyancy is an upward force equal to half its weight. Drag from the water can be modeled by F = kv, where k = 0.650 kg/s.

(a) Determine the terminal speed of the sinking rock.

(b) Determine its depth, speed, and acceleration 1.50 seconds after it is released.

(c) At what depth will it be at 99.0% of its terminal speed?


Homework Equations



Net force (vertical), F = ma, F = kv

The Attempt at a Solution



The solution to a) was easy enough.

Fnet = mg + FB + Fdrag

Fnet = -3.92 N + 1.96 N + Fdrag = 0 N (at terminal speed)

Fdrag = 1.96 N = 0.650 kg/s *vt

vt = 3.02m/s.

b) For calculating the depth, speed, and acceleration, I think I need to integrate the function of the net force with respect to time to get the equations for velocity and position. But I have not had much luck doing so because I am rusty on my integration.

The equation for the acceleration is

a(t) = -4.90 N + 1.63s-1 *v(t)

so I'm thinking

dv/dt = -4.90 N + 1.63s-1 *dx/dt

but then I get something like

dv = -4.90 N dt + 1.63s-1 *dx

and I don't know what to do with dx?

c) I'm thinking the answer for this part will be easy enough once I have the answer to b...
 
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  • #2
bump:smile:
 
  • #3
ultimateman said:
The equation for the acceleration is

a(t) = -4.90 N + 1.63s-1 *v(t)

so I'm thinking

dv/dt = -4.90 N + 1.63s-1 *dx/dt
Instead of dx/dt, try leaving that as v in this equation.
 

What is terminal velocity?

Terminal velocity is the maximum velocity that an object can reach while falling through a fluid, such as air or water. It occurs when the force of gravity on the object is equal to the force of air resistance acting on the object.

How is terminal velocity calculated?

The formula for calculating terminal velocity is Vt = √(2mg/ρAC), where Vt is terminal velocity, m is the mass of the object, g is the acceleration due to gravity, ρ is the density of the fluid, A is the cross-sectional area of the object, and C is the drag coefficient.

What is buoyancy?

Buoyancy is the upward force exerted on an object immersed in a fluid. It is caused by the pressure difference on the top and bottom of the object, with the greater pressure on the bottom resulting in a net upward force.

How is buoyant force calculated?

The formula for calculating buoyant force is Fb = ρVg, where Fb is the buoyant force, ρ is the density of the fluid, V is the volume of the displaced fluid, and g is the acceleration due to gravity.

What is the relationship between F, k, and v in the equation F = kv?

In the equation F = kv, F represents the force acting on the object, k is a constant that depends on the properties of the fluid and the size and shape of the object, and v is the velocity of the object. This equation shows how the force acting on an object in a fluid is directly proportional to its velocity.

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