Work and Kinematics: Do Both Stones Hit the Ground at the Same Time and Speed?

[dhphysics]

Homework Statement

A stone is thrown straight up from the roof of a building with initial speed v0. At the same instant, a second stone is thrown from the roof upward at an angle of 60o with the horizontal and with the same initial speed v0. Which is correct?
both stones hit the ground at the same time and with equal speeds.
the stones hit the ground at different times but with equal speeds.
The stones hit the ground at different times and with different speeds.
both stones hit the ground at the same time but with different speeds.

Homework Equations

vf^2 = vo^2 + 2*a*x
x = vo*t + 1/2*a*t^2

The Attempt at a Solution

For the stone that is straight up, the time from when the stone is thrown to when it gets to the top can be found by using the formula vf^2 = vo^2 + 2*a*x, if vf is 0. Therefore, the time t = v0/g

For the stone thrown at an angle, the y component of the v0 is v0sin60. By using the same formula, the time t = v0sin60/g.

V0sin60/g is smaller than v0/g, so the stone thrown at an angle takes longer to hit the ground. That means the stones hit the ground at different times.

The speeds are also not the same because the v0 for the two stones are not the same.

However, this answer is not correct. Could you point out where the mistake in my reasoning is?

Thanks

 

[phinds]

I must be missing something (or your answer book is wrong) since I agree that they HAVE to hit the ground at different times and with different speeds. The problem statement seems very clear but are you SURE you have transcribed it exactly?

 

[dhphysics]

I have transcribed it exactly (but it should read 60 degrees instead of 60o). The problem has something to do with Work and Energy, but I'm not sure how to apply those ideas to the problem.

 

[NTW]

The stones hit the ground at different times, because the initial velocity is the same in both cases, but the vertical component is different.

And they hit the ground with the same velocity, as there's no friction in this scenary, and kinetic energy is conserved...

 

[phinds]

And they hit the ground with the same velocity, as there's no friction in this scenary, and kinetic energy is conserved...

How can they hit the ground at the same velocity since one goes higher than the other before starting down?

 

[NTW]

How can they hit the ground at the same velocity since one goes higher than the other before starting down?

The initial KE was the same in both cases when the stones were launched. When returning to the ground, that KE can't have varied, as there is no friction and the masses of the stones are the same... Hence, the strike velocity is the same...

 

[phinds]

The initial KE was the same in both cases when the stones were launched. When returning to the ground, that KE can't have varied, as there is no friction and the masses of the stones are the same... Hence, the strike velocity is the same...

OK, I get it. My mistake was that I was ONLY considering the vertical component of the velocities. That don't hit the ground with the same VERTICAL velocity, but that's not the whole picture. Thanks for clarifying that.

 

[nasu]

As they are thrown from the roof and fall to the ground the final kinetic energy is not the same as the initial KE.
However their final KEs (and so their speeds) are the same. The work done by gravity is the same in both cases.

A kinematic calculation gives the same result, of course.
For the first one,
$v^2_1f=v_0^2+2gh$
and for the second one
$v^2_2f=(v_0^2 sin^2 60^o+2gh)+(v_0^2 cos^2 60^o)$

where h is the height of the roof.

 

[NTW]

Well, nowhere was stated the roof was at any height above the ground, so I took it as zero. After all, the vertically-launched stone would have landed where it was shot, and the problem stated that both stones 'hit the ground'. Hence, there was a valid reason to consider the roof and the ground level... There are (a few) buildings that are like that, after all...