Comparing Mechanical Energies of two falling masses

[rustyshackle]

Ok, first time posting...hopefully I do this correctly, as I have found myself in a bind on this one. It is a question comparing the mechanical energy of two masses falling in different ways in which you are given the option of greater than, less than, or equal to.

Homework Statement

The figure below (In the attachment) shows two blocks, m1 and m2 of the same mass. Block 1 starts from rest and slides down a frictionless incline from a height H from the floor while block 2 is dropped from rest at the same time from the same height.At height h block 1 falls off the incline.

1) The velocity of block 1 when it is at height h is (greater than/less than/equal to) the velocity of block2 when it is also at height h.

2) The kinetic energy of block 2 when it is at height his (greater than/less than/equal to) the kinetic energy of block 1 when it is also at height h.

3) The change in potential energy of block 1 when it slides from H to h is (greater than/less than/equal to) the change in potential energy of block 2 when it falls from H to h.

4) The time it takes for block 1 to fall from h to the floor is (greater than/less than/equal to) the time it takes block 2 to fall from h to the floor.

5) The time it takes for block 2 to fall from H to h is (greater than/less than/equal to) the time it takes to slide the same vertical distance.

Homework Equations

PE = mgh
KE = .5mv^2
v= sqrt(2gh)

The Attempt at a Solution

My attempts so far I have been incorrect, but here is what I thought it was and why:

1) Equal to. Because both have the same change in potential energy and by energy conservation principle will have the same kinetic energy at the bottom of the slide
when there is no friction. Both will have speed v = sqrt(2gh)

2) Equal to. Both have the same height at h and therefore have equal PE, meaning equal PE.

3) Equal to. While the change is more sudden in mass two, both are undergoing the same change in PE from H to h.

4) Equal to. Once they reach point h, they have the same speed and same height, and therefore reach it at the same time.

5) Greater than. Mass 2 has a more sudden change in PE and there gets up to speed faster than mass 1.

Any help on this would be greatly appreciated, as I have referenced the internet, my textbook, and lecture notes and have still not found any help. Thank you.

 

[azizlwl]

1. less than block#2. Block#1 has horizontal and vertical acceleration. Block#2 only vertical acceleration.
2. Equal. It is conservative force. Total work done does not depend on path.
3. Equal as above.
4 Greater than. As (1) less initial vertical velocity.
5. Less than. Higher acceleration than sliding

 

[rustyshackle]

Thanks a lot, 1-3 seem to be correct, but I just realized I incorrectly posted #4 and #5 they should read (edited above):

4) The time it takes for block 1 to fall from h to the floor is (greater than/less than/equal to) the time it takes block 2 to fall from h to the floor.

So wouldn't this be less than since block two had the larger vertical acceleration during its initial free fall?

5) The time it takes for block 2 to fall from H to h is (greater than/less than/equal to) the time it takes to slide the same vertical distance.

I think this one is less than as the PE change is much more sudden in block 2.

Thanks again.

 

[azizlwl]

Thanks a lot, 1-3 seem to be correct, but I just realized I incorrectly posted #4 and #5 they should read (edited above):

4) The time it takes for block 1 to fall from h to the floor is (greater than/less than/equal to) the time it takes block 2 to fall from h to the floor.

So wouldn't this be less than since block two had the larger vertical acceleration during its initial free fall?

5) The time it takes for block 2 to fall from H to h is (greater than/less than/equal to) the time it takes to slide the same vertical distance.

I think this one is less than as the PE change is much more sudden in block 2.

Thanks again.

question 4
You are right,at h they have different initial velocity.
Block 1 has less intial vertical velocity, thus slower than block 2 to reach the floor. Thus the time is greater.
Both have same vertical acceleration due to gravity.


For question 5,
Verticall acceleration equal to g
If it slide the acceleration will be equal to gSinθ.
For equal distance the time for sliding must be greater due to less acceleration.

 

[cheekujodhpur]

Hey, people! I am still doubtful to the answer of the first question.
The Block#1 after falling through the same distance must have a velocity of the same magnitude, whatever may be the direction of acceleration. This is validated by the fact that gain in kinetic energy must be equal to the loss in potential energy. And since, you stated yourself that change in P.E. is same, then change in K.E. must also be same, since the total energy is conserved.

 

[azizlwl]

Hey, people! I am still doubtful to the answer of the first question.
The Block#1 after falling through the same distance must have a velocity of the same magnitude, whatever may be the direction of acceleration. This is validated by the fact that gain in kinetic energy must be equal to the loss in potential energy. And since, you stated yourself that change in P.E. is same, then change in K.E. must also be same, since the total energy is conserved.

We are comparing the velocity of block 2 to block 1
Block 2 has only one velocity downward.
Block 1 has 2 components of velocity, vertical which we compare to block 2 and horizontal velocity.
You can add KE of horizontal and KE of vertical, and then you get total KE which is equal to block 2.
Energy is a scalar unit.