Calculating velocity of object (final) down a ramp.

[physicsluvme]

Hi physicsforums mates,

I came across an interesting application question online on optimal mass / velocity.
The context was given that an object on wheels have to be released down a ramp of angle θ (assume to be anything). It has to travel a far distance after reaching the end of the ramp and have a good buoyancy on water.

May I know equations of dynamics and forces to apply in this scenario to achieve the optimum mass?

Thank you.

 

[haruspex]

Much more information needed.
Does the ramp suddenly level out or smoothly?
What considerations bring it to rest - air resistance, rolling resistance?
What has the water to do with it? Does it have to travel a long way skimming over water, or is this just a limit on the density?

In general, the denser and more aerodynamic you can make it the further it will go.

 

[physicsluvme]

Yup it's basically a triangular ramp with a specific angle and specific height. Say the angle of elevation is 20degrees and vertical height is 1meter.

For water it just has to be as buoyant as possible on a separate note, not related to the first ramp scenario. Thanks!

 

[mfb]

Now you give two criteria at the same time (rolling distance and buoyancy), without any metric to compare them.

Make it large and aerodynamic, try to reduce friction at the wheels and let the total density be smaller than 1g/cm^3.
Your problem has no real optimum with the conditions you posted here.

 

[haruspex]

Now you give two criteria at the same time (rolling distance and buoyancy), without any metric to compare them.

Make it large and aerodynamic, try to reduce friction at the wheels and let the total density be smaller than 1g/cm^3.
Your problem has no real optimum with the conditions you posted here.

I agree. There's no constraint given which limits the size, only the density. Since air resistance relates to area, the bigger the better.
The sudden change in slope presents a challenge. When the object hits the bottom of the ramp there will be an impact, wasting energy. To minimise this, it helps to have a lot of the energy stored in angular momentum rather than linear, so make it a big ring.

 

[physicsluvme]

Sorry for not making it clear but the dimension limits is half a metre for length, height as well as breadth.
The point is to be as buoyant as possible but at the same time be able to travel far distances.

Essentially here are some of my questions
1) Does it mean that if I have a heavier object, it'll have more GPE and hence more KE and hence travel further when it reach the bottom of the ramp?
2) If the above is true and I have to balance the distance with buoyancy what is the best approach to achieve the magical number for the optimum mass ?

 

[mfb]

1) Does it mean that if I have a heavier object, it'll have more GPE and hence more KE and hence travel further when it reach the bottom of the ramp?

No, as its friction on the ground will be larger by the same amount. However, air friction does not depend on the mass (it mainly depends on the surface area and shape), therefore you can reduce the friction/mass ratio a bit with higher mass.

The point is to be as buoyant as possible but at the same time be able to travel far distances.

This is still not well-defined.
Consider solid object A which rolls 5 meters and has an average density of 0.5g/cm^3 (which means that half of its volume will stick out of the water). Now compare it with solid object B which rolls 8 meters and has an average density of 0.7g/cm^3 (which means that about 30% of it will stick out of the water).

Which one is better?

Even "as buoyant as possible" is not clear: What about designs like usual ships? They have no problem to swim in the correct orientation, and most of its volume can be above the water line. But if you tilt them too much, they will sink. Compare this to water ice: It will swim in every possible orientation, but most of its volume is below the water line.

 

[haruspex]

Given the limits on dimensions, I now recommend a solid cylinder. By storing much of the KE in rotational form you minimise the losses both wrt the impact at the bottom of the ramp and in air resistance. If allowed a sealed cavity, maybe a hollow cylinder with the central hole closed at the ends.
Regarding the impact, it may help to use a rubbery material.
If the length of the slope is prescribed but you can choose the angle, there will be an optimum one. If the height is prescribed, make the slope as long and shallow as you can.

 

[physicsluvme]

I see, thanks for the advices.
Am I right to say that since Kinetic Friction = KFC x N, Kinectic Friction is directly proportionate to Normal Force which is then directly proportionate to weight.
Since then, comparing MGH and KFCxN, if I increase the mass of the object, would MGH increase more than the increase of KFCxN and hence giving me a higher net force down the ramp?

Sorry if I sound vague, I'm trying to get a better concept of dynamics.

 

[haruspex]

You'll get a higher net force, but only in proportion to the mass, and the rolling resistance beyond that is also proportional to the mass, so no improvement in distance traveled.
As mentioned, the advantage of a higher mass is because air resistance depends on area, not mass.
Btw, should distinguish between friction and rolling resistance, which a lot of people confuse. With a wheeled vehicle, the drag from the axles is friction; the drag from the wheels on the road is rolling resistance (the energy required to flex the rubber).