Solving Physics Project: Initial Velocity of Dart?

[Crupler22]

Hello and thanks to anyone in advance. So I have this physics project that me and a partner have to do. We are suposed to construct a lab for the situation he gives us. Ours is: a dart hits a block hanging on a string which causes it to reach a max height. So we are suposed to figure out the initial velocity of the dart by finding the height it reached. So I started writing an equation. It goes like this:

I first used the momentum equation.
m1v1 + m2v2= (m1+m2)v3 -so the blocks not moving so you can take the second part out.
m1v1=(m1+m2)v3 - divided both sides by (m1+m2) to get v3 alone, so

m1v1
-------- = v3
(m1+m2)

Then science all the energy is sent in a circular path I substatuted v3 into the centripetal Force equation. Fc=(mv^2)/r so

mass that's rotating-->(m1+m2) * ((m1v1)/(m1+m2))^2r so stuff cancels, and I get:
(m1v1)^2/(m1+m2)r
k so that's force so now I figure since its traveling a distance its work (W=F*D)
D is going to be the arc length of the circular path that it takes, so its (Theta* r)
W= (m1^2)(v1^2)
---------------- * theta r = W so the r's cancel
(m1+m2)r

and then I realize all this kenetic energy that is done when its at its top height is equal to the potenial energy. so

(m1+m2)gh = (m1^2)(v1^2)
--------------- *Theta
(m1+m2)
so then I solve for V1 which is the inital velocity of the dart.

v1 = srq( (m1+m2)^2gh
------------------
m1^2 * theta )
But my teacher said that he never saw that before and said there is an easier way which I know now. But is this right?

thanks

 

[berkeman]

The last part using PE and KE would be the way I'd do it. I'm not sure your math is correct, though (hard for me to read).

 

[Doc Al]

I first used the momentum equation.
m1v1 + m2v2= (m1+m2)v3 -so the blocks not moving so you can take the second part out.
m1v1=(m1+m2)v3 - divided both sides by (m1+m2) to get v3 alone, so

m1v1
-------- = v3
(m1+m2)

This part's good.

Then science all the energy is sent in a circular path I substatuted v3 into the centripetal Force equation. Fc=(mv^2)/r so

mass that's rotating-->(m1+m2) * ((m1v1)/(m1+m2))^2r so stuff cancels, and I get:
(m1v1)^2/(m1+m2)r

So you found the centripetal force acting on the system when it's at the bottom. Note that this force changes as the system moves up and that it acts perpendicular to the direction of motion.

k so that's force so now I figure since its traveling a distance its work (W=F*D)
D is going to be the arc length of the circular path that it takes, so its (Theta* r)
W= (m1^2)(v1^2)
---------------- * theta r = W so the r's cancel
(m1+m2)r

This doesn't make sense. Since the force and displacement are at right angles to each other, the force certainly doesn't do any work. (The force doing the real work is gravity, which acts downward.)

and then I realize all this kenetic energy that is done when its at its top height is equal to the potenial energy. so

(m1+m2)gh = (m1^2)(v1^2)
--------------- *Theta
(m1+m2)
so then I solve for V1 which is the inital velocity of the dart.

v1 = srq( (m1+m2)^2gh
------------------
m1^2 * theta )
But my teacher said that he never saw that before and said there is an easier way which I know now. But is this right?

It's certainly true that energy is conserved, so KE at the bottom equals the PE at the top. But, as I pointed out, your method is incorrect even if it happened to give you a correct answer. But the answer's not correct. Compare it to the answer you'd get by setting KE = PE (not "W" = PE).

 

[Crupler22]

Ooooooooh Ok, I get it. Since the motion of the mass always wants to travel in a path perpendicular to the circular path, because of inertia. And since the force is centripetal its acting inward, so their perpendicular. So just curious, if I then did something to make the distance relative to the force it would work? Like since there at right angles use some trig function to set them in the same direction? I know the real equation I was just curious if I could get mine to work. Thanks Doc Al

 

[Doc Al]

Here's what would work that is kind of like what you were doing. Find the force in the direction of motion. There are two forces acting on the system at any point: the string tension and the weight. The tension is always perpendicular to the motion, so that contributes no work. But the weight will have a component in the direction of motion as the system rises up in an arc, so it does do work on the system. But since that component is a function of angle (it's not constant) you can't just multiply by the arc length--you have to integrate.