An astronaut on a spaceship traveling at 0.75c relative to Earth measures his ship to be 25 m long. On the ship, he eats his lunch in 23 min. What length is the spaceship according to observers on Earth?

The system of equations are given as

$$\begin{array}{r} x+y+2 z=a \\ x+z=b \\ 2 x+y+3 z=c \end{array}$$

Verify that for this system to be consistent, the constants $a$, $b$, and $c$ must satisfy $c=a+b$.

The length of a spaceship is measured to be exactly half its proper length. a) What is the speed of the spaceship relative to the observer's frame? b) What is the dilation of the spaceship's unit time?

The rest radius of the Earth is $6370$ km and its orbital speed about the Sun is $29.8$ km/s. By how much would the Earth’s diameter appear to be shortened to an observer stationed so as to be able to watch the Earth move past at this speed?

An electron moves to the right with a speed of 0.90c relative to the laboratory frame. A proton moves to the right with a speed of 0.70c relative to the electron. Find the speed of the proton relative to the laboratory frame.

Movement of the chromosomes during anaphase would be most affected by a drug that prevents _____.

Name the stage of meiosis during which chromatids are separated to opposite poles of the cell.

In a nuclear reaction two identical particles are created, traveling in opposite directions. If the speed of each particle is 0.75 c, relative to the laboratory frame of reference, what is one particle's speed relative to the other particle?

When two moles of hydrogen and one mole of oxygen react to form two moles of water, the energy released is 484 kJ. How much does the mass of the elements decrease in this reaction? What % of the total original mass of the system does this mass change represent?

On their 21st birthday, one twin gets on a moving sidewalk, which carries her out to star X at speed $\dfrac{4}{5}c$; her twin brother stays home. When the traveling twin gets out to star X, she immediately jumps onto the returning sidewalk and comes back to earth, again at speed $\dfrac{4}{5}c$. She arrives on her 39th birthday (as determined by her watch). (a) How old is her twin brother? (b) How far away is star X? (Give your answer in light years.) Call the outbound sidewalk system $\tilde{\mathcal S}$ and the inbound one $\overline{\mathcal S}$ (the earth system is $\mathcal S$). All three system choose their coordinates and their master clocks such that $x=\overline x=\tilde x=0$, $t=\overline t=\tilde t$ at the moment of departure. (c) What are the coordinates (x,t) of the jump (from outbound to inbound sidewalk) in $\mathcal S$? (d) What are the coordinates ($\overline x,\overline t$) of the jump in $\overline{\mathcal S}$? (e) What are the coordinates ($\tilde x, \tilde t$) of the jump in $\tilde{\mathcal S}$? (f) If the traveling twin wants her watch to agree with the clock in $\tilde{\mathcal S}$, how must she reset it immediately after the jump? What does her watch then read when she gets home? (This wouldn't change her age, of course--she's still 39--it would just make her watch agree with the standard synchronization in $\tilde{\mathcal S}$.) (g) If the traveling twin is asked the question, "How old is your brother right now?", what is the correct reply (i) just before she makes the jump, (ii) just after she makes the jump? (Nothing dramatic happens to her brother during the split second between (i) and (ii), of course; what does change abruptly is his sister's notion of what "right now, back home" means.? (h) How many earth years does the return trip take? Add this to (ii) from (g) to determine how old she expects him to be at their reunion. Compare your answer to (a).

A record turntable of radius R rotates at angular velocity $\omega$. The circumference is presumably Lorentz-contracted, but the radius (being perpendicular to the velocity) is not. What's the ratio of the circumference to the diameter, in terms of $\omega$ and R? According to the rules of ordinary geometry, it has to be $\pi$. What's going on here?

A farm boy studying physics believes that he can fit a 15.0-m-long pole into a 12.0-m-long barn if he runs fast enough (carrying the pole). Can he do it? Explain in detail. How does this fit with the idea that when he is running the barn looks even shorter to him than 12.0 m?

A rocket ship leaves the earth at a speed of $\dfrac{3}{5}c$. When a clock on the rocket says 1 hour has elapsed, the rocket ship sends a light signal back to earth. (a) According to earth clocks, when was the signal sent? (b) According to earth clocks, how long after the rocket left did the signal arrive back on earth? (c) According to the rocket observer, how long after the rocket left did the signal arrive back on earth?

An electron is accelerated so that its kinetic energy is greater than its rest energy $m_{0} c^{2}$ by a factor of 999. What is the speed of the electron in each case?