Metal block sliding horizontally

In summary, a metal block of mass m sliding on a horizontal surface with a viscous resistance that varies as the 3/2 power of speed cannot travel farther than 2mvo^(1/2)/c, where vo is the initial speed of the block at x = 0. This can be shown by integrating the equation F = ma = m(dv/dt) = -cv^(3/2) and solving for the distance traveled, which is equal to 2m/c times the square root of the initial speed vo.
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Homework Statement


A metal block of mass m slides on a horizontal surface that has been lubricated
with a heavy oil so that the block suffers a viscous resistance that varies as the 3/2
power of the speed: F(v) = -cv3/2

If the initial speed of the block is vo at x = 0, show that the block cannot travel farther than 2mvo1/2/c

Homework Equations


F = ma = m[itex]\frac{dv}{dt}[/itex]

The Attempt at a Solution


So, I took

m[itex]\frac{dv}{dt}[/itex] = -cv3/2
I rearranged it and got

dv/v3/2= [itex]\frac{-c}{m}[/itex]dt

I've tried integrating this and I can't seem to end up with the right answer. I'm totally lost. Anyone have any ideas?

EDIT: No help at all?
 
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  • #2
You are not integrating wrt the correct variables; you need dx, not dt.

Use the chain rule to write dv/dt = dv/dx dx/dt = dv/dx v.

Then when you rearrange the terms you will be integrating dv/sqrt(v) = -c/m dx
The integrals run (v0, v_final) and (0, d); if we carry the integral to v_final = 0 then the distance covered will be d_final.

When I do this I get 2m/c sqrt(v0) = d_final, which supports the expected conclusion.
 
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What is the concept of a metal block sliding horizontally?

The concept refers to the movement of a metal block, which is placed on a smooth horizontal surface, from one point to another.

What factors affect the sliding motion of a metal block?

The sliding motion of a metal block can be affected by various factors such as the surface roughness, the weight of the block, the force applied, and the presence of any external forces like friction or air resistance.

How do scientists study the sliding motion of metal blocks?

Scientists use experiments, mathematical models, and simulations to study the sliding motion of metal blocks. They also use various instruments such as force sensors, motion detectors, and high-speed cameras to collect data and analyze the motion.

What are the applications of studying metal block sliding horizontally?

Studying metal block sliding horizontally has various applications in industries such as manufacturing, transportation, and construction. It also helps in understanding the principles of motion and friction, which have implications in fields like engineering and physics.

How can the sliding motion of a metal block be optimized?

The sliding motion of a metal block can be optimized by reducing the surface roughness, controlling the weight and force applied, and minimizing the effects of external forces such as friction. Understanding the principles of motion and using lubricants can also help in optimizing the sliding motion.

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