# Tutor profile: Marta R.

## Questions

### Subject: Astrophysics

A $$30M_\odot$$ star reaches the end of its life and explodes. A $$10M_\odot$$ black hole is left, and the rest of the star is ejected by the explosion. If the total explosion energy is $$10^{46}J$$ and 10% is converted to the kinetic nergy of the ejecta. What is the initial expansion speed of the supernova remnant after the explosion?

To solve this problem, we need to set the variables. The ejected mass ($$m_{E}$$) is $$20M_\odot$$ once the star was originally $$30M_\odot$$ and the black hole is $$10M_\odot$$. The efficiency of the explosion ($$ \epsilon$$) is 10% and the supernova energy ($$E_{SN}$$) is $$10^{46}J$$. With the Kinetic Energy formula and calling the initial velocity $$V_{EJ}$$, the formula becomes: $$\frac{1}{2}m_{E}V_{EJ}^{2}=\epsilon \times E_{SN} $$ and rearranging the equation, $$V_{EJ}=({\frac{2\times E_{SN} }{m_{E}} )}^{\frac{1}{2}}$$. Converting from solar masses to KG, (1$$M_\odot$$=2\times 10^{30}KG) we get $$V_{EJ}=7.1\times 10^{3} kms^{−1}$$.

### Subject: Physics

A car is travelling at $$15 ms^{-1}$$. The car accelerates uniformly to $$33 ms^{-1}$$. The car's acceleration is $$1.2 ms^{-2}$$. What is the distance covered while the car is accelerating?

This problem is solved using S.U.V.A.T. We have $$u=15 ms^{-1}$$, $$v=33 ms^{-1}$$ and$$a=1.2 ms^{-2}$$. The formula we can use is: $$v^{2}=u^{2}+2as$$. Rearraging this, we get, $$s=\frac{v^{2}-u^{2}}{2a}$$. So, $$s=\frac{33^{2}-15^{2}}{2\times1.2}$$=360m. The car covered a 360m distance while accelerating.

### Subject: Partial Differential Equations

Find u(x,y) from the following $$ u_{y} = x^{2}+y^{2}$$

To find this, we have to first integrate both x and y with respect to y and as it is $$u_{y}$$ there will be a function of x (I will call it f(x)) that will not appear in the above eqaution but has to be accounted for when constructing u(x,y). So, integrating $$x^{2}$$ with respect to y, we get $$x^{2} y$$ and integrating $$y^{2}$$ with respect to y, we get $$ \frac{y^{2}}{3} $$. So $$u(x,y)=x^{2} y+\frac{y^{2}}{3}+f(x)$$ where f(x) is the function of x that is not accounted for in $$u_{y}$$

## Contact tutor

needs and Marta will reply soon.