Physics 131-01 Final Exam

I pledge that I have neither given nor received unauthorized assistance during the completion of this work.

Signature height0pt depth1pt width3in

Questions (4 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. You are riding on a flat surface in a cart at a velocity $\vec v_{launcher}\hat i$. A stationary observer is nearby. At $t=0$ your coordinate systems coincide. At $t>0$ you fire the toy cannon at an angle of about $45^\circ$ from the moving cart. Consider a point ${\vec r} = x{\hat i} + y{\hat j}$ on the ball's trajectory in the stationary observer's reference frame. If the moving observer's time frame is moving at the speed then what would the moving observer measure for $x$? Call this horizontal position of the moving observer $x'$.
   
   
   
   
   
   
   

2. Consider the following scenario from the Twins Paradox of Special Relativity. As a space-faring twin's craft recedes from the Earth it is moving at a constant speed. Since no inertial frame can be considered ``better'' than any other there is nothing physically inconsistent with the view that the space-faring twin is observing the Earth recede from her at a constant velocity. Hence, the space-faring twin will observe clocks on the Earth to move slowly and the Earth-bound twin will age at a slower rate than the space-faring one. Is this reasoning flawed? Explain.
   
   
   
   
   
   
   

3. How does the vertical acceleration of a projectile on the Earth influence the horizontal acceleration? What is your evidence?
   
   
   
   
   
   
   

4. In the figure below, what is the equation to estimate the average slope of the curve at the highlighted point in terms of , , , and ? How would you find the ``exact'' value of the slope at the point in the figure between $t_1$ and $t_2$?

   

   
   
   

5. Consider the histogram shown below for a set of possible fall times for an experiment studying the acceleration of gravity. What advantages does a histogram of these data have over calculating the average and standard deviation?

   
   
   

   

   
   
   

6. You are sleeping at home over winter break and wake up to find the house is on fire and smoke is pouring into the partially open bedroom door. The room is so messy that you cannot get to the door. The only way to close the door is to throw either a blob of clay or a super ball at the door -- there's not enough time to throw both. Assuming that the clay blob and the super ball have the same mass, which would you throw to close the door: the clay blob (which will stick to the door) or the super ball (which will bounce back with almost the same velocity it had before it collided with the door)? Give reasons for your choice, using any notions you already have or any new concepts developed in physics such as force, momentum, Newton's laws, etc. Remember, your life depends on it!
   
   
   
   
   
   
   

7. An object can move along a horizontal line (the + position axis). Assume that friction is so small that it can be ignored. The object's velocity-time graph is shown below.

   
   
   

   
   
   
   

   Draw the the shape of the acceleration-time and force-time graphs on the axes below. Explain your reasoning.

   
   
   

   
   
   
   
   
   
   
   
   
   

8. Suppose that the force applied to the object in Question 7 were twice as large. Sketch with dashed lines on the same axes above the force, acceleration, and velocity. Explain your reasoning.
   
   
   
   
   
   
   

9. The figure below shows three apples that are launched from the same level with the same speed. One moves straight upward, one is launched at a small angle to the vertical, and one is launched along a frictionless incline. Rank the apples according to their speed when they reach the level of the dashed line, greatest first. Explain.

   

   
   
   
   

10. In some motorcycle races, the riders drive over small hills and become airborne for short periods of time. If the motorcycle racer keeps the throttle open while leaving the hill and going into the air, the motorcycle tends to nose upward. Why?

    
    
    
    
    
    
    
    
    

Problems. Clearly show all reasoning for full credit. Use a separate sheet for your work.

1.	10 pts.	A supertrain (proper length $l_p=120~m$) travels at a speed $v=0.90c$ as it passes through a tunnel of proper length $l_t = 50~m$. As seen by a trackside observer, is the train ever completely within the tunnel? If so, with how much space to spare?
2.	10 pts.	The average speed of a nitrogen molecule in air is about $v_N= 6.5\times 10^2~m/s$ and its mass is $m_N = 4.68\times 10^{-26}~kg$. If it takes a time $\Delta t = 3.5\times 10^{-13}~s$ for a nitrogen molecule to hit a wall and rebound with the same speed, but moving in the opposite direction, what is the average acceleration $a_N$ of the molecule during this time interval? What average force $F_N$ does the molecule exert on the wall?

Continue $\rightarrow$

Problems (continued). Clearly show all work for full credit.

3.	10 pts.	A student throws a set of keys vertically upward to her sorority sister who is in a window a distance $h=3~m$ above. The keys are caught a time $t_1=1.0~s$ later by the sister's outstretched hand. What was the initial velocity $v_0$ of the keys? What was the velocity $v_1$ of the keys just before they were caught?
4.	10 pts.	A student sits on a freely rotating stool holding two weights, each of mass $m_w=4~kg$. When his arms are extended horizontally, the weights are a distance $l_0=1.1~m$ from the axis of rotation and he rotates with an angular velocity $\omega_i = 0.8~rad/s$. The moment of inertia of the student plus stool is $I_{s}=3.5~kg/m^2$ and is assumed to be constant throughout this problem. The student pulls the weights inward to a distance $l_1 = 0.3~m$ from the axis of rotation. Treat the weights as point particles. What is the student's new angular velocity $\omega_f$? What is the kinetic energy before and after he pulls the weights inward?
5.	10 pts.	A car of mass $m_1 = 1400~ kg$ heading north and moving at $v_1 = 35~ mph$ collides in a perfectly inelastic collision with a truck of mass $m_2 = 4000~ kg$ going East at $v_2 = 20~mph$. What percentage of the total mechanical energy is lost from the collision?

Continue $\rightarrow$

Problems (continued). Clearly show all work for full credit.

6.

10 pts.

The stars on the rim of our Milky Way Galaxy take about 750 million years (i.e. $T = 7.5\times 10^8 ~ yrs=2.4\times 10^{16}~s$) to orbit the galactic core which is a distance $r_G = 9.5 \times 10^{20}~ m$ away. They follow circular orbits. What is the acceleration of these rim stars? What galactic mass is required? Assume, as usual, the mass of the galaxy can be treated as a point at the center of the galaxy. How does this compare to the galactic mass of $1.4\times 10^{41}~kg$ determined by counting the number of stars?

Table of Constants

Speed of Light ($c$)	$2.9979\times 10^8~m/s$	proton/neutron mass	$1.67\times 10^{-27}~kg$
$R$	$8.31J/K-mole$	$g$	$9.8~m/s^2$
Gravitation constant	$6.67 \times 10^{-11}~N-m^2/kg^2$	Electron mass	$9.11\times 10^{-31}~kg$
Earth's mass	$5.98 \times 10^{24}~kg$	Earth's radius	$6.37\times 10^6~m$
Earth-Moon distance	$3.84\times 10^8~m$	Earth-Sun distance	$1.496\times 10^{11}~m$
Solar mass	$1.991\times 10^{30}kg$	Solar radius	$6.96\times 10^{8}~m$

Table of Equations

$$\Delta \vec r = \vec r_{finish} - \vec r_{start} \quad \Delta \vec v = \vec v_{finish} - \vec v_{start}$$

$$\langle \vec v \rangle = {\Delta \vec r \over \Delta t} \qqu... ...ta t \to 0} {\vec r (t+\Delta t) - \vec r(t) \over \Delta t}$$

$$\langle \vec a \rangle = {\Delta \vec v \over \Delta t} \qqu... ...lta t \to 0} {\vec v(t+\Delta t) - \vec v(t) \over \Delta t}$$

$$x(t) = {1 \over 2}at^2 + v_0t + y_0 \quad v = at + v_0 \quad... ...er r} \quad ( \vec v \perp \vec r \quad \vec v \perp \vec a_c)$$

$$\vec F_{net} = \sum_i \vec F_i = m \vec a = {d \vec p \over dt} \qquad \vec F_{AB} = -\vec F_{BA}$$

$$\vert\vec F_f\vert = \mu N \quad \vert\vec F_c\vert = m {v^2... ...} \quad \vec F_s(x) = -kx\hat i \quad \vec F_g(y) = -mg\hat j$$

$$W = \int \vec F \cdot d\vec s = \int \vert\vec F\vert ~\ve... ... KE = {1 \over 2} mv^2 \quad KE_{rot} = {1 \over 2} I \omega^2$$

$$KE_0 + U_0 = KE_1 + U_1 \quad U_s(x) = {1 \over 2} kx^2 \quad U_g(y) = mgy \quad U_G = \frac{G m_1 m_2}{r}$$

$$\vec p = m \vec v \quad \vec p_0 = \vec p_1 \quad \vec r_{cm} = \frac{\sum \vec r_i m_i}{\sum m_i}$$

$$\theta = {s \over r} \quad \omega = {v_T \over r} = {d\theta... ...ega \over dt} \quad I = \sum m_i r_i^2 = I_{cm} + Mh^2 \quad$$

$$\vert\vec \tau \vert = r F\sin\phi = I\alpha = \left \vert\f... ...mv\sin \phi = I \omega \quad L_0 = L_1 \quad v_{cm} = r\omega$$

$$\Delta t = {\Delta t_p \over \sqrt {1 - {v^2 \over c^2}}} \q... ...^\prime = x - vt \quad y^\prime = y \quad v_i^\prime = v_i - v$$

$$\vec A = A_x \hat i + A_y \hat j + A_z \hat k \qquad {dA \ov... ... 0 \qquad {dt \over dt} = 1 \qquad {dt^2 \over dt} = 2t \qquad$$

$$\int f(x)dx = \lim_{\Delta x \rightarrow 0} \sum f(x_i) \D... ... x \qquad \int dx = x + c \qquad \int x dx = {x^2 \over 2} + c$$

$$\sin \theta = {opp \over hyp} \quad \cos \theta = {adj \over... ...over adj} \quad x^2 + y^2 + z^2 = R^2 \quad \rho = {m \over V}$$

$$x = {-b \pm \sqrt{b^2 -4ac} \over 2a} \quad C = 2 \pi r \qua... ...pi r^2 \quad Area = {1 \over 2}bh \quad Area = 4 \pi r^2 \quad$$