Answer:
![v_f = 15 \frac{m}{s}](https://tex.z-dn.net/?f=%20v_f%20%3D%2015%20%5Cfrac%7Bm%7D%7Bs%7D%20%20)
Explanation:
We can solve this problem using conservation of angular momentum.
The angular momentum
is
![\vec{L} = \vec{r} \times \vec{p}](https://tex.z-dn.net/?f=%5Cvec%7BL%7D%20%20%3D%20%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bp%7D)
where
is the position and
the linear momentum.
We also know that the torque is
![\vec{\tau} = \frac{d\vec{L}}{dt} = \frac{d}{dt} ( \vec{r} \times \vec{p} )](https://tex.z-dn.net/?f=%5Cvec%7B%5Ctau%7D%20%3D%20%5Cfrac%7Bd%5Cvec%7BL%7D%7D%7Bdt%7D%20%20%3D%20%5Cfrac%7Bd%7D%7Bdt%7D%20%28%20%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bp%7D%20%29)
![\vec{\tau} = \frac{d}{dt} \vec{r} \times \vec{p} + \vec{r} \times \frac{d}{dt} \vec{p}](https://tex.z-dn.net/?f=%5Cvec%7B%5Ctau%7D%20%3D%20%20%5Cfrac%7Bd%7D%7Bdt%7D%20%20%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bp%7D%20%2B%20%20%20%5Cvec%7Br%7D%20%5Ctimes%20%5Cfrac%7Bd%7D%7Bdt%7D%20%5Cvec%7Bp%7D%20)
![\vec{\tau} = \vec{v} \times \vec{p} + \vec{r} \times \vec{F}](https://tex.z-dn.net/?f=%5Cvec%7B%5Ctau%7D%20%3D%20%20%5Cvec%7Bv%7D%20%5Ctimes%20%5Cvec%7Bp%7D%20%2B%20%20%20%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7BF%7D%20)
but, as the linear momentum is
this means that is parallel to the velocity, and the first term must equal zero
![\vec{v} \times \vec{p}=0](https://tex.z-dn.net/?f=%5Cvec%7Bv%7D%20%5Ctimes%20%5Cvec%7Bp%7D%3D0)
so
![\vec{\tau} = \vec{r} \times \vec{F}](https://tex.z-dn.net/?f=%5Cvec%7B%5Ctau%7D%20%3D%20%20%20%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7BF%7D%20)
But, as the only horizontal force is the tension of the string, the force must be parallel to the vector position measured from the vertical rod, so
![\vec{\tau}_{rod} = 0](https://tex.z-dn.net/?f=%5Cvec%7B%5Ctau%7D_%7Brod%7D%20%3D%20%20%200%20)
this means, for the angular momentum measure from the rod:
![\frac{d\vec{L}_{rod}}{dt} = 0](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5Cvec%7BL%7D_%7Brod%7D%7D%7Bdt%7D%20%3D%20%20%200%20)
that means :
![\vec{L}_{rod} = constant](https://tex.z-dn.net/?f=%5Cvec%7BL%7D_%7Brod%7D%20%3D%20constant)
So, the magnitude of initial angular momentum is :
![| \vec{L}_{rod_i} | = |\vec{r}_i||\vec{p}_i| cos(\theta)](https://tex.z-dn.net/?f=%7C%20%5Cvec%7BL%7D_%7Brod_i%7D%20%7C%20%3D%20%7C%5Cvec%7Br%7D_i%7C%7C%5Cvec%7Bp%7D_i%7C%20cos%28%5Ctheta%29)
but the angle is 90°, so:
![| \vec{L}_{rod_i} | = |\vec{r}_i||\vec{p}_i|](https://tex.z-dn.net/?f=%7C%20%5Cvec%7BL%7D_%7Brod_i%7D%20%7C%20%3D%20%7C%5Cvec%7Br%7D_i%7C%7C%5Cvec%7Bp%7D_i%7C%20)
![| \vec{L}_{rod_i} | = r_i * m * v_i](https://tex.z-dn.net/?f=%7C%20%5Cvec%7BL%7D_%7Brod_i%7D%20%7C%20%3D%20r_i%20%2A%20m%20%2A%20v_i)
We know that the distance to the rod is 0.750 m, the mass 2.00 kg and the speed 5 m/s, so:
![| \vec{L}_{rod_i} | = 0.750 \ m \ 2.00 \ kg \ 5 \ \frac{m}{s}](https://tex.z-dn.net/?f=%7C%20%5Cvec%7BL%7D_%7Brod_i%7D%20%7C%20%3D%200.750%20%5C%20m%20%5C%202.00%20%5C%20kg%20%5C%205%20%5C%20%5Cfrac%7Bm%7D%7Bs%7D%20)
![| \vec{L}_{rod_i} | = 7.5 \frac{kg m^2}{s}](https://tex.z-dn.net/?f=%7C%20%5Cvec%7BL%7D_%7Brod_i%7D%20%7C%20%3D%207.5%20%5Cfrac%7Bkg%20m%5E2%7D%7Bs%7D%20)
For our final angular momentum we have:
![| \vec{L}_{rod_f} | = r_f * m * v_f](https://tex.z-dn.net/?f=%7C%20%5Cvec%7BL%7D_%7Brod_f%7D%20%7C%20%3D%20r_f%20%2A%20m%20%2A%20v_f)
and the radius is 0.250 m and the mass is 2.00 kg
![| \vec{L}_{rod_f} | = 0.250 m * 2.00 kg * v_f](https://tex.z-dn.net/?f=%7C%20%5Cvec%7BL%7D_%7Brod_f%7D%20%7C%20%3D%200.250%20m%20%2A%202.00%20kg%20%2A%20v_f%20)
but, as the angular momentum is constant, this must be equal to the initial angular momentum
![7.5 \frac{kg m^2}{s} = 0.250 m * 2.00 kg * v_f](https://tex.z-dn.net/?f=%207.5%20%5Cfrac%7Bkg%20m%5E2%7D%7Bs%7D%20%3D%200.250%20m%20%2A%202.00%20kg%20%2A%20v_f%20)
![v_f = \frac{7.5 \frac{kg m^2}{s}}{ 0.250 m * 2.00 kg}](https://tex.z-dn.net/?f=%20v_f%20%3D%20%5Cfrac%7B7.5%20%5Cfrac%7Bkg%20m%5E2%7D%7Bs%7D%7D%7B%200.250%20m%20%2A%202.00%20kg%7D%20)
![v_f = 15 \frac{m}{s}](https://tex.z-dn.net/?f=%20v_f%20%3D%2015%20%5Cfrac%7Bm%7D%7Bs%7D%20%20)