# Do different length ramps violate conservation of energy?

• Aperture
In summary: Sorry, I am still not clear on this. Did the original problem ask about the final speed or the time taken?
Aperture
Homework Statement
Say you have 2 different lamps that start and end at the same heights. One ramp is steeper and longer than the other, while the second ramp is the shortest line from the top to the bottom. If you released two identical balls, one on each ramp, from rest, you would find that the ball on the steeper ramp has a greater final velocity. How is this possible if mgh=1/2m(v^2) and m, g, and h are the same for both cases?
Relevant Equations
mgh=(1/2)(m)(v^2)
mgh=(1/2)(m)(v^2)
gh=(1/2)v^2
sqrt(2gh)=v
Should have the same v, but this is not the case based on the answer and real-life experiments.

When are you measuring the final velocity? At the end of the ramp, or once the ball is rolling horizontally after leaving the ramp?

Drakkith said:
When are you measuring the final velocity? At the end of the ramp, or once the ball is rolling horizontally after leaving the ramp?
end of the ramp, when h = 0 so only 1/2mv^2 remains.

Aperture said:
Homework Statement: Say you have 2 different lamps that start and end at the same heights. One ramp is steeper and longer than the other, while the second ramp is the shortest line from the top to the bottom. If you released two identical balls, one on each ramp, from rest, you would find that the ball on the steeper ramp has a greater final velocity. How is this possible if mgh=1/2m(v^2) and m, g, and h are the same for both cases?
Relevant Equations: mgh=(1/2)(m)(v^2)

mgh=(1/2)(m)(v^2)
gh=(1/2)v^2
sqrt(2gh)=v
Should have the same v, but this is not the case based on the answer and real-life experiments.
Can you show the two ramps in a drawing? I cannot imagine a ramp that is steeper and longer than the other if they start and end at the same heights.

Also, why do you say "If you released two identical balls, one on each ramp, from rest, you would find that the ball on the steeper ramp has a greater final velocity." Did you do an experiment?

On Edit: Here is a convincing demonstration that should clarify what's going on. Be sure to watch the video in its entirety.

Last edited:
DrClaude, Aperture, Lnewqban and 1 other person
kuruman said:
why do you say "If you released two identical balls, one on each ramp, from rest, you would find that the ball on the steeper ramp has a greater final velocity."
The way I read post #1, the OP is there quoting the given answer. Indeed, the OP believes they should end with the same speed.
Of course, if the nonlinear path is always below the linear one then, ignoring losses, the ball will arrive sooner, though at the same speed.
If we take losses into account, but no slipping, then the straight line should produce the greatest final speed.
Edit: I need to check that; it's not as obvious as I thought.

If the gradient is gentle until a final descent too steep to maintain rolling then that may give the greatest final speed.

Last edited:
Yes, this involves a combination of rolling and slipping. I missed that this is a homework problem posed as an energy non-conservation paradox.

kuruman said:
Can you show the two ramps in a drawing? I cannot imagine a ramp that is steeper and longer than the other if they start and end at the same heights.

Also, why do you say "If you released two identical balls, one on each ramp, from rest, you would find that the ball on the steeper ramp has a greater final velocity." Did you do an experiment?

On Edit: Here is a convincing demonstration that should clarify what's going on. Be sure to watch the video in its entirety.

The video cleared things up. Thanks.

Drakkith
Aperture said:
The video cleared things up. Thanks.
Well that's great, but can you please still clarify why you said this:
Aperture said:
Should have the same v, but this is not the case based on the answer and real-life experiments.

Was it just a misunderstanding of some data or something? Thanks.

berkeman said:
Well that's great, but can you please still clarify why you said this:Was it just a misunderstanding of some data or something? Thanks.
Yes. I failed to consider that a faster total time does not mean a faster final speed.

Drakkith
Aperture said:
Yes. I failed to consider that a faster total time does not mean a faster final speed.
Sorry, I am still not clear on this. Did the original problem ask about the final speed or the time taken?

haruspex said:
The way I read post #1, the OP is there quoting the given answer. Indeed, the OP believes they should end with the same speed.
Of course, if the nonlinear path is always below the linear one then, ignoring losses, the ball will arrive sooner, though at the same speed.
If we take losses into account, but no slipping, then the straight line should produce the greatest final speed.
Edit: I need to check that; it's not as obvious as I thought.

If the gradient is gentle until a final descent too steep to maintain rolling then that may give the greatest final speed.
Fwiw, I considered descending in two straight lines, first at angle ##\theta_1##, by ##h_1## vertically, then at ##\theta_2##, by ##h_2## vertically. Drag per unit mass is ##kv^2##.
I get a final speed given by ##\alpha_2v^2=1-e^{-2\alpha_2gh_2}(1-(\frac{\alpha_2}{\alpha_1})(1-e^{-2\alpha_1gh_1}))##, where ##h_1+h_2=Y##, ##h_1\cot(\theta_1)+h_2\cot(\theta_2)=X## and ##\alpha_i=k/(g\sin(\theta_i))##.
Maximisation wrt ##h_1, \theta_1## would be the next step, but I might not get to it.

An alternative, of course, is the full calculus of variations approach. Haven't tried it.

## Do different length ramps violate conservation of energy?

No, different length ramps do not violate the conservation of energy. The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. The total mechanical energy (potential + kinetic) remains constant if only conservative forces are doing work.

## How does the length of a ramp affect the energy of an object moving down it?

The length of the ramp affects the manner in which potential energy is converted to kinetic energy. A longer ramp results in a more gradual conversion, but the total energy at the bottom of the ramp will be the same, assuming no energy is lost to friction or air resistance.

## Why does an object take longer to travel down a longer ramp if energy is conserved?

An object takes longer to travel down a longer ramp because the ramp provides a more gradual slope, reducing the acceleration due to gravity along the ramp's surface. The total energy remains conserved, but the time taken to convert potential energy to kinetic energy is extended.

## Does friction affect the conservation of energy on different length ramps?

Yes, friction affects the conservation of energy. If friction is present, some of the mechanical energy is converted into thermal energy, meaning the total mechanical energy (potential + kinetic) at the bottom of the ramp will be less than it would be in a frictionless scenario. However, the total energy, including thermal energy, remains conserved.

## Can different ramp lengths result in different final speeds for an object?

In the absence of friction and air resistance, different ramp lengths will not result in different final speeds for an object. The final speed depends only on the vertical height from which the object descends. With friction or air resistance, the final speeds can differ due to energy losses.

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