Final velocities with the inclusion of air resistance

[sohjau]

Homework Statement

Two rocks are thrown off a cliff at the same initial speed, v. The first rock is launched at an angle, ϴ, directed below the horizontal. The second rock is launched above the horizontal at the same angle. If air resistance is negligible, the rocks would hit the ground at the same final speed. Including air resistance, which rock would have a greater final speed? Solve using a conservation of energy argument.

Homework Equations

The Attempt at a Solution

If you make the angle 90°, you only have to think about the vertical components. Ignoring air resistance, rock 2 would reach the same speed as it was launched as once it reaches its original altitude. That would result in both the rocks having the same final speed. Including air resistance, rock 2 would not reach as high due to the drag force and would also not reach its initial speed at its original altitude. Therefore, rock 1 would have the greater final speed. Would that be the case for every angle? If the cliff is high enough, won't the rocks reach terminal velocity and hit the ground at the same final speed as well? How does one solve this using a conservation of energy argument?

 

[kuruman]

How does one solve this using a conservation of energy argument?

Just consider which path dissipates more energy.

 

[haruspex]

If the cliff is high enough, won't the rocks reach terminal velocity

Terminal velocity is never quite reached, in theory.

Just consider which path dissipates more energy.

Although it is intuitively obvious, a rigorous proof is not that simple. How would you prove which path dissipates more? The horizontal component of speed complicates things. The whole trajectory is different.

 

[kuruman]

Although it is intuitively obvious, a rigorous proof is not that simple.

I agree, but is there an argument that makes it possible for air resistance to dissipate less energy over the longer path?

 

[haruspex]

I agree, but is there an argument that makes it possible for air resistance to dissipate less energy over the longer path?

Inability to argue that it could happen does not constitute proof it could not.