Which is Faster, Going Up or Coming Down?

[Zashmar]

Homework Statement

Suppose you throw a 1kg ball into the air. Determine whether the ball takes
longer to reach its maximum height or fall back to Earth from its maximum
height?

Assume the forces acting on the ball are the force of gravity and a retarding
force of air resistance with direction opposite to the direction of motion.
(-mkv) (assume k = 0.1)


Would it not just me the same? How would I show this

PS. This is not being used for an assignment

 

[LCKurtz]

Homework Statement

Suppose you throw a 1kg ball into the air. Determine whether the ball takes
longer to reach its maximum height or fall back to Earth from its maximum
height?

Assume the forces acting on the ball are the force of gravity and a retarding
force of air resistance with direction opposite to the direction of motion.
(-mkv) (assume k = 0.1)


Would it not just me the same? How would I show this

PS. This is not being used for an assignment

My intuition says they would be the same too, but that might be wrong. To prove it you need to set up the DE and solve it.

 

[Dick]

My intuition says they would be the same too, but that might be wrong. To prove it you need to set up the DE and solve it.

I don't think so. There's an intuitive way to deal with this. Suppose you are at a height h above where you started. Do you have more or less kinetic energy coming up than going down? What does that mean about velocity going up versus coming down?

 

[Zashmar]

I'm not sure...that there is less kinetic upwards that coming down? How would one algebraically show this?

 

[Dick]

I'm not sure...that there is less kinetic upwards that coming down? How would one algebraically show this?

You don't have to. The air resistance is always taking energy away because it's opposite to the direction of motion. Do you know about kinetic and potential energy?

 

[Zashmar]

Do you know about kinetic and potential energy?

Not really, I know an object has gravitational potential energy?

 

[Ray Vickson]

I don't think so. There's an intuitive way to deal with this. Suppose you are at a height h above where you started. Do you have more or less kinetic energy coming up than going down? What does that mean about velocity going up versus coming down?

I don't think you can rely on conservation of energy in this problem. The ball's total energy is not necessarily conserved, because frictional forces may be dissipative, and some energy may be transferred to the atmosphere in the form of heat, etc.

 

[LCKurtz]

My intuition says they would be the same too, but that might be wrong. To prove it you need to set up the DE and solve it.

I don't think so. There's an intuitive way to deal with this. Suppose you are at a height h above where you started. Do you have more or less kinetic energy coming up than going down? What does that mean about velocity going up versus coming down?

Agreed. I wasn't thinking in those terms. Solving the DE would be the hard way.

 

[Feodalherren]

Can't you just use conservation of energy? (in case this is for physics and not math)
Show that the potential energy when the ball is at max h = the kinetic energy right before the ball hits the ground.

 

[LCKurtz]

I don't think you can rely on conservation of energy in this problem. The ball's total energy is not necessarily conserved, because frictional forces may be dissipative, and some energy may be transferred to the atmosphere in the form of heat, etc.

Can't you just use conservation of energy? (in case this is for physics and not math)
Show that the potential energy when the ball is at max h = the kinetic energy right before the ball hits the ground.

Dick's point is that energy is lost. That gives some energy inequalities.

 

[Feodalherren]

Hm fair enough. I don't know then. It's beyond my capabilities.

 

[Zashmar]

Does anyone know how I could algebraically prove this?

 

[LCKurtz]

Does anyone know how I could algebraically prove this?

Sure. Like I said in post #2, set up the DE and solve it. Once you have the solution you can figure out how long it takes to hit the max height and how much additional time it takes to hit the ground. That will leave the intuition out of it. In the DE give it an initial height of $0$ and an initial velocity of $v_0$. Or take $v_0=1$, it shouldn't matter.

 

[HallsofIvy]

But the whole point is that the total kinetic energy plus potential energy is NOT conserved since you lose some energy to air resistance. At any given height, the potential energy is the same so coming down, the kinetic energy, and so the speed, must be less than it was going up.

 

[D H]

That was Dick's point in post #3.

 

[Ray Vickson]

Does anyone know how I could algebraically prove this?

It is impossible to tell which message you are now responding to, and what you mean by 'this' (as in 'prove this algebraically'). To avoid this in future please respond by using the 'quote' button.

Anyway, on the upward part both forces act against the direction of motion; that is, forces add. On the way down, one force acts in the direction of motion and the other acts in the opposite direction, so there is a partial cancellation of forces on the downward trip. This suggests that the upward trip decelerates faster than the downward trip accelerates, so the up-trip should take less time than the down-trip. You can verify this explicitly in numerical examples, by solving the differential equations and solving some transcendental equations that arise from the DE solutions.