# Tutor profile: Stephanie B.

## Questions

### Subject: Applied Mathematics

Father Christmas' sled sits on a slope at an angle $$\theta=arctan(3/4)$$ degrees to the horizontal. If the sled weights $$500 kg$$, and the coefficient of friction between the sleigh and the snow is $$0.2$$, how much force will the reindeer have to exert parallel to the slope in order to just get the sled to move?

The horizontal force, parallel to the snow is given by $$F_H=m*g*sin(\theta) +0.2*m*g*cos(\theta)$$ $$F_H=500*g*3/5+100*g*4/5 $$ where $$g=9.81ms^{-2}$$ is the gravitational acceleration on the Earth. $$F_H=380g=3727.8 N$$

### Subject: Calculus

What is the most general form of a quadratic function? How would you calculate the gradient of the function $$m$$ at some general point $$(P,Q)$$ on the graph?

The general form of a quadratic function is $$f(x)=ax^2+bx+c$$ and the gradient of the function at a general point $$(P,Q)$$ is $$\frac{df(x)}{dx}=2ax+b$$ at $$x=P$$ : $$m=2aP+b$$

### Subject: Physics

Derive an equation, using the principle of conservation of energy, for the escape velocity of a rocket of mass m leaving the surface}of a planet of mass $$M$$ and radius $$r_o$$. Define any constants you use and give the units of this escape velocity.

The principle of conservation of energy tells us that the potential energy of the rocket on the surface of the planet is equal to the minimum kinetic energy it needs to escape the gravitational pull of the planet. $$KE=PE$$ $$ \rightarrow \frac{1}{2}mv_{esc}^2= \frac{GMm}{r_o}$$ $$ \rightarrow v_{esc}=\sqrt{\frac{2GM}{r_o}}$$ $$G$$ is Newton's constant and $$v_{esc}$$ has units meters per second $$(ms^{-1})$$

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