# Fall 2011 Midterm 1 Solutions

## Midterm 1 Solutions - Q1 (Ave Score: 22.3/30)

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?

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

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

$v^{2}=0.2g$

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

## Midterm 1 Solutions - Q2 (Ave Score: 26.5/35)

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?

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