Here I'm collecting some spacetime-smart, metric-first, or traveler-kinematic puzzlers that might be useful in this class or that. The math for all of this stuff is quite simple, does not even require calculus if you are willing to take given relationships for granted, and involves equations patterned after those used in regular introductory physics courses.
The main idea new to traditional introductions is that time is local to the clock that is measuring it. This gives rise to a number of new variables, and surprising connections (like that between energy and differential aging).
Volunteers to put together detailed sample-solutions e.g. in the extended captions to the attached figures are welcome to share their approaches as well as their questions/comments.
Forces on the way to school: You are passenger in a car accelerating at 0.5 gee from rest at a stop sign. If your speed-change (within the accelerated frame of your car) is zero, then what backwards geometric-acceleration (blue in this animation) on every ounce of your being appears (in that accelerated frame) to be canceling out the forward proper-acceleration caused by the horizontal force on you from the back of your seat?
Forces rounding the curve: You are in a car going at constant speed v around a leftward curve of radius R. In order to hold your velocity-change (within the accelerated frame of your car) to zero, what rightward (radially-outward) geometric-acceleration on every ounce of your being (blue in this animation) must appear (in that accelerated frame) to cancel out the centripetal proper-acceleration caused by the leftward (radially-inward) force of your car seat on you? Also in which direction would the resulting torque appear to operate, if your center of mass is located above your point of contact with the seat?
Loop the loop: You are pilot of an inverted aircraft going 200 mph at the top of a half-mile radius loop-the-loop (illustrated in a different setting with this animation). What net geometric-force toward the cupholder is the coffee feeling in your now-inverted cup?
Super highways: A car's clock reads 10:00:00.000 as it passes the first bank's clock which is also reading 10:00:00.000, but then reads 10:00:00.999 as it passes the second bank's clock when it is reading 10:00:01.000 instead. (a) How fast is the car going assuming that the two bank clocks were synchronized. (b) How far apart are the two banks if those time-readings list hours:minutes:seconds, so that the car's clock is now a millisecond behind? (Caution: this distance may be quite large.)
High speed-limit legislation: On a car trip with 300 map-miles remaining, (a) is it proper-velocity or coordinate-velocity which determines most directly how much longer (on your clock) that you have to remain in the vehicle? (b) Is it proper-velocity or coordinate-velocity which determines most directly what time you can be expected to arrive on clocks at your destination? (c) What upper limit on vehicle momentum, vehicle kinetic energy, driver reaction-time for obstacle avoidance, and pedestrian reaction-time for oncoming-vehicle avoidance, is provided by a proper-speed limit of w/c in the range of positive real numbers? (d) How would these relationships change for a coordinate-speed limit of v/c instead?
Mr. Tompkin's bike trip around town: Read the first chapter of George Gamow's book about Mr. Tompkins, and see if you can estimate the speed of light in his dream.
Land speed records: (a) What's the proper-velocity (w ≡ dx/dτ = γv) land-speed-record for protons accelerated in the laboratory on earth? How about for (b) electrons, or for (c) iron nuclei? How do these values compare to (d) the corresponding land-speed records stated in terms of coordinate-velocity (v ≡ dx/dt)?
The cosmic-ray example: Muons with a lifetime of Δτ = 50 [micro-seconds] are created at Δx = 100 [km] altitude, in collisions between air molecules and high-energy cosmic rays. Since these muons routinely make it to sea level for detection before their on-board clocks cause them to decay, how fast (earth distance traveled per unit earth time) must they be moving?
A fast airtrack: The timer carried by a high speed airtrack glider registers only one third of the elapsed time recorded by gate clocks, on the trip between a pair of synchronized gates fixed to the airtrack. How fast was the glider traveling?
Bank clocks revisited: The highly-accurate timer carried by a fast-moving vehicle registers only 99% of the elapsed-time recorded by synchronized bank clocks that it passes. How fast was the vehicle traveling?
Land-speed collision-records: What is the land speed record in terms of relative proper-velocity for the collision of (a) two electrons? How about for (b) two protons? The good news here, of course, that the cost of making a collider might not be much more than twice that of making an ordinary acceleration. Also, what are these same land-speed collision-records stated in terms of (c) their relative coordinate velocities?
Deep-space confrontation: A starfleet battle-cruiser drops out of hyperspace in the orbital plane of a ringworld, traveling at 1 lightyear/traveler-year ≈ 0.707 lightyear/map-year radially away from the ringworld's star. An enemy cruiser drops out of hyperspace nearby at the same time, traveling 1 ly/ty in the rotation-direction of the ringworld's orbit, and in a direction perpendicular to the starfleet cruiser's radial-trajectory. What is the proper-velocity (magnitude and direction) of the enemy cruiser relative to the starfleet ship?
The kick of a ray-gun: How does the recoil of a U = 1.8[MJ] laser pulse (grey dot high up on the curtain) with the momentum of a speeding automobile of comparable energy.
PET scan puzzler: The atomic nucleus of a radio-isotope that you just ingested emits an anti-matter electron, which rather quickly finds an electron to annihilate to create a pair of photons heading in opposite directions. What is (roughly) the energy, frequency, momentum, and wavelength of each of these photons?
The jump to lightspeed: Can you (a) estimate the energy it would take to accelerate you (not counting your spaceship) up to the relativistic proper-speed of one lightyear per traveler year? This is an optimum speed if we are mainly concerned about time-elapsed not on traveler but on map clocks, since speed-increases beyond this mainly decrease traveler not map time-elapsed during a trip. The kinetic energy acquired on this “jump to lightspeed” is equivalent to (b) how many gallons of gasoline, each of which contains 100 million Joules of energy available to do work?
Millennium Falcon puzzler: Your 1000 kg space-roadster accelerates from rest to 1 [lightyear/traveler-year] i.e. ~0.707 [lightseconds/map-second] in a few seconds on the big screen. (a) The kinetic energy acquired on this “jump to lightspeed” is equivalent to how many gallons of gasoline, each of which contains 100 million Joules of energy available to do work? (b) The momentum acquired per unit time of course should give you some idea of the force (and divided by mass the acceleration) involved as well. Even if you had the necessary fuel, do you really want this jump to happen that fast?
Failed high-speed get-away: An enemy spaceship with its FTL-drive disabled drops out of hyperspace in the neighborhood of a ringworld habitat, traveling at 1 ly/ty radially away from the ringworld's star. A starfleet battle-cruiser capable of continuous 1 gee acceleration at rest nearby takes up the chase. (a) How much cruiser time and enemy time elapses before the cruiser can catch up to the enemy? (b) What are the ringworld coordinates and time for that encounter? (c) What is the relative proper-speed of the two ships when that encounter occurs?
Accelerated twin puzzler: A twin leaves her sibling on earth for a 10 lightyear jaunt in which she accelerates at 1 "gee" for 5 lightyears distance, decelerates through another 5 lightyears to a stop, accelerates in the earthward direction for another 5 lightyears, and then decelerates to a half back on earth. How much older are she and her stay-at-home twin at the end of this trip?
Inter-stellar shuttle: How much (a) traveler-time Δτ would elapse on a 1-gee constant proper-acceleration roundtrip to and from Sirius if it were 8.6 lightyears away from Earth? How much (b) map-time Δt would elapse on the same trip? How much (c) on-board fuel would be required at the start of each one-way leg of the trip, assuming that the fuel could be efficiently converted to photons directed in the desired thrust direction.
Inter-galactic adventure: What's (a) the elapsed-time on traveler-clocks for a 1-gee constant proper-acceleration round-trip to Andromeda galaxy say 2 million lightyears away? How much time (b) will elapse on map-time (e.g. earth or milkyway-mean-time i.e. MMT) clocks during that trip? As far as actually doing this, for 100% efficient matter-antimatter photon-propulsion (c) what's the minimum launchmass/payload ratio for a one-way trip? Finally, (d) what's the minimum launchmass/payload ratio for a round trip i.e. where you didn't plan to refuel at your destination point before the return trip?
GPS accuracy: If the reported clock-time on a GPS satellite orbiting at 5 earth-radii was not corrected (a) for gravitational time-dilation by your GPS system, how much distance-error (time-error times lightspeed) would build up over a one-minute "check-in" interval? By how much (b) would the effects of satellite-motion reduce the need for this correction? Also (c) at what orbital radius might there be no need for a correction at all?
Head versus feet ages: Imagine that standing on the earth's surface is like undergoing an upward but constant proper-acceleration for an extended period of time. What estimate would this give for the differential aging between your head and your feet as a 1.8 meter tall adult, if you spent 25% of your life standing during the last 10 years?
A geological anomaly: If the official formation-date for the earth was 4.5 billion years ago, how much older than the earth's center is its surface today?