Which ball travels further, up a ramp or thrown in the air

[Alexstrasza]

Homework Statement

Two identical balls, one is thrown at angle α up a frictionless surface, the other one is thrown at the same angle up in the air.

Both have the same initial velocity.

A. Which one travels further? Explain.
B. Is the mechanical energy of the first ball conserved? Explain.

Homework Equations

F=ma
K(e)=mv^2/2, P(e)=m*g*h
Momentum... p=m*v
W=Fa

The Attempt at a Solution

[/B]
A. I am having trouble with this one.

I tried to use v(f)=v(i)+2ax but then it is the same equation for both the balls. That can't be right?

I think that the ball in the air would travel the shorter distance, because it doesn't have the support of the ramp.

B. The energy of the first ball is conserved, because there is no friction and the only force except gravity working on that ball is N, which doesn't do work because it's perpendicular.

 

[mfb]

I tried to use v(f)=v(i)+2ax but then it is the same equation for both the balls. That can't be right?

What are the accelerations for the two balls? Are they the same?
What is x, and how does that equation work if you have to consider two dimensions?

I think that the ball in the air would travel the shorter distance, because it doesn't have the support of the ramp.

There is also a different effect in the opposite direction.

B. The energy of the first ball is conserved, because there is no friction and the only force except gravity working on that ball is N, which doesn't do work because it's perpendicular.

Is it really perpendicular?
What exactly counts as mechanical energy?

 

[Doc Al]

I'm not sure what is meant by "travel further". Horizontal distance? Height?

 

[J Hann]

The "Range" equation will quickly give you the distance the ball in air will travel.
Now if the initial kinetic energy is converted to potential energy,
how far can the ball travel on the incline?

 

[Alexstrasza]

Thanks for the replies everyone. Sorry for late update.

The question was asking to find which ball will reach the maximum height. I tried to solve using energy conservation but then it gives me the same solution for both.

Is it really perpendicular?
What exactly counts as mechanical energy?

Isn't N always perpendicular to the surface? Energy = initial velocity squared x mass / 2 and energy is conserved.

The "Range" equation will quickly give you the distance the ball in air will travel.
Now if the initial kinetic energy is converted to potential energy,
how far can the ball travel on the incline?

That is what I am confused about, what is the difference between how far the ball travels up an incline vs. at an angle in the air. I am pretty sure that the incline "helps" the ball but not sure how to express it.

 

[Doc Al]

The question was asking to find which ball will reach the maximum height.

Ah, now the question makes sense.

I tried to solve using energy conservation but then it gives me the same solution for both.

Describe what you did for each. They are not quite the same.

Hint: What's the speed of each ball when it reaches maximum height?

 

[J Hann]

As Doc Al implied, consider the "total" energy at the top of the trajectory for each case.

 

[mfb]

I tried to solve using energy conservation but then it gives me the same solution for both.

Energy is not the only conserved quantity that is relevant for the freely falling ball.

Isn't N always perpendicular to the surface?

The force from the surface is, the force from gravity is not.

 

[SammyS]

The "Range" equation will quickly give you the distance the ball in air will travel.
Now if the initial kinetic energy is converted to potential energy,
how far can the ball travel on the incline?

No. It will not give distance traveled.

It will only give the displacement.

 

[haruspex]

B. The energy of the first ball is conserved, because there is no friction and the only force except gravity working on that ball is N, which doesn't do work because it's perpendicular.

Quite so, the normal force does no work. But forces acting on an object can transfer the energy of an object from one form to another without doing any net work. E.g. when a ball rolls down a slope, the friction transfers energy into rotational KE.
This is the way that the ramp 'helps' the ball go higher.

 

[SammyS]

Frictionless surface.

 

[haruspex]

Frictionless surface.

Yes, I understand that. I was just using rolling down a frictional slope as an example of how a force that does no work can transfer energy from one mode to another. The same happens here, but not in respect of rotation.