Kinetic Energy and Potential Energy

[gkangelexa]

Two stones are thrown with the same initial speed at the same instant from the roof of a building. One stone is thrown at an angle of 30 degree above the horizontal; the other is thrown horizontally.(Neglect air resistance.), which of the following is true:

(a)The stones strike the ground at the same time and with equal speeds.
(b)The stones strike the ground at the same time with different speeds.
(c)The stones strike the ground at different times with equal speeds.
(d)The stones strike the ground at different times with different speed.



The answer is c. I understand why they arrive at different times. But I don't understand why they would arrive with equal speeds because the first stone was thrown 30 degrees higher than the second stone so wouldn't it have a higher U= mgh and thus a faster velocity?

 

[Pengwuino]

The answer is c. I understand why they arrive at different times. But I don't understand why they would arrive with equal speeds because the first stone was thrown 30 degrees higher than the second stone so wouldn't it have a higher U= mgh and thus a faster velocity?

No, initially the objects have the exact same kinetic and potential energy because they are both thrown with the same kinetic energy and at the same heights. Conservation of energy tells us that if they both have the same energy at the beginning, they'll have the same energy at the ground if they both land at the same height.

Remember, if the object is fired upwards, it will be continuously exchanging kinetic energy for potential energy thus losing speed as it goes higher and higher until it reaches the top of it's trajectory and then begins falling. At that point, it begins exchanging potential energy for kinetic energy and begins gaining speed.

 

[Drakkith]

This means that the increase in vertical velocity is equal in energy to the other stones horizontal velocity when they hit the ground?

 

[Physicsnuubie]

Hi.
I have to first say that I am not a teacher.

I can't explain this in terms of energies. but, this problem can also be explained using kinematics.

When the stone is being thrown 30 degrees upwards at initial velocity, u, at the roof of the building, the velocity deceases until 0 ms^-1 at the highest point the stone can reach and then the stone will start to fall downwards. when the stone reaches back at the height of the roof of the building, the velocity of the stone is back to u ms^-1. Just that this time round, the stone is falling downwards instead of going upwards.

and if you can see, both the stone actually have the same velocity of u ms^-1 at the same height in the same direction. this explains why both of the stones will arrive the ground at equal speed.

Just that the stone which was threw upwards will take a longer time to reach the ground as it has to travel upwards first before coming down.

I hope you can understand... :)

 

[gkangelexa]

The part where I'm confused involves the initial vertical velocities.

The stone that was thrown horizontally has 0 initial vertical velocity, whereas the part that was thrown upward 30 degrees has an initial velocity component.

This means that at the same height that you referred to, the first stone had 0 vertical velocity whereas the second stone will have a vertical velocity component.

 

[Drakkith]

Since the question used "speed", would the horizontal velocity can?

 

[Pengwuino]

The part where I'm confused involves the initial vertical velocities.

The stone that was thrown horizontally has 0 initial vertical velocity, whereas the part that was thrown upward 30 degrees has an initial velocity component.

This means that at the same height that you referred to, the first stone had 0 vertical velocity whereas the second stone will have a vertical velocity component.

Yes, that's true. Let's assume the initial velocity was 10m/s. The object thrown horizontally would have a horizontal component of 10m/s and 0m/s in the vertical. However, the object thrown at an angle of 30 degrees will have a horizontal component of 10m/s*cos(30) and a vertical component of 10m/s*sin(30). They both have the same speed, but the object shot off at an angle will have a horizontal component less than 10m/s.

 

[gdbb]

It took me a couple minutes to justify this in my head. The problem I had at first was that I was stuck thinking only about the y-component of the velocities. At the instant before each stone hits the ground, their velocity vectors will have the same magnitude (which is the definition of speed), but differing directions. One's vector might be 45 degrees below the horizontal and the other's might be 60 degrees below the horizontal, but the magnitudes of each will be the same.