**Question 1.** (30 points) A smooth block of mass 100g is sliding along the edge of a smooth cone with constant speed. The height of the cone is 20cm, and half of it's apex angle is 30$^{o}$.

**A.** (5 points) Draw a free body diagram which represents all the forces acting on the block.

**B.** (5 points) What is the magnitude of the gravitational force acting on the block?

$mg=0.1\mathrm{kg}\times9.81\mathrm{ms^{-2}}=0.981\mathrm{N}$

**C.** (5 points) What is the magnitude of the component of the gravitational force on the block which points down the slope of the cone?

$mg\sin(60^{o})=0.85N$

**D.** (5 points) What is the magnitude of the normal force acting on the block?

$F_{N}\sin(30^{o})=mg$

$F_{N}=\frac{0.981\mathrm{ms^{-2}}}{0.5}=1.962\mathrm{N}$

**E.** (10 points) What is the speed of the block?

$\frac{mv^{2}}{r}=F_{N}\cos(30^{o})=\frac{mg}{\tan(30^{o})}$

$r=0.2\tan{30^{o}}$

$v^{2}=0.2g$

$v=1.4\mathrm{ms^{-1}}$

**Question 2.** (35 points) A plane is flying horizontally with a constant speed of 100m/s at a height $h$ above the ground, and drops a 50kg bomb with the intention of hitting a car that has just begun driving up a 10$^{o}$ incline which starts a distance $l$ in front of the plane. The speed of the car is a constant 30m/s. For the following questions use the coordinate axes defined in the figure, where the origin is taken to be the initial position of the car. (Note: The car has been stolen by a Martian trying to get hands on experience with our GPS system and our planet's survival depends on us stopping the Martian).

**A.** (5 points) What is the initial velocity of the bomb relative to the car? Write your answer in unit vector notation.

$v_{x}=100-30\cos(10^{o})=70.46\mathrm{ms^{-1}}$

$v_{y}=-30\sin(10^{o})=-5.21\mathrm{ms^{-1}}$

$\vec{v}=70.46\mathrm{ms^{-1}}\hat{i}-5.21\mathrm{ms^{-1}}\hat{j}$

**B.** (5 points) Write equations for both components ($x$ and $y$) of the car's displacement as a function of time, taking t=0s to be the time the bomb is released.

$x=30\cos(10^{o})t=29.54t\,\mathrm{m}$

$y=30\sin(10^{o})t=5.21t\,\mathrm{m}$

**C.** (5 points) Write equations for both components ($x$ and $y$) of the bomb's displacement as a function of time, taking t=0s to be the time the bomb is released.

$x=100t-l\,\mathrm{m}$

$y=h-\frac{1}{2}gt^{2}\,\mathrm{m}$

**D.** (5 points) If the bomb hits the car at time t=10s what was the height of the plane above the ground $h$ when it dropped the bomb?

$y=52.1\mathrm{m}$

$52.1=h-\frac{1}{2}g10^{2}$

$h=52.1+50\times9.81=542.61\mathrm{m}$

**E.** (5 points) What is the horizontal displacement of the plane relative to the car when the bomb hits the car at t=10s.

$0\mathrm{m}$

**F.** (5 points) How much work did gravity do on the bomb while it was falling?

$mg\frac{1}{2}g10^{2}=50\times50\times9.81^2=240590\mathrm{J}$

**G.** (5 points) How much kinetic energy does the bomb have when it hits the car?

$\frac{1}{2}mv_{0}^{2}+240590=\frac{1}{2}\times50\times100^{2}+240590=250000+240590=490590\mathrm{J}$