Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• It is well known that the speed of sound in air is u = 330 m/s at standard temperature and pressure. What this means is that sound travels at speed u in all directions in the frame S where the air is at rest. In any other frame S′, moving relative to S, its speed is not u in all directions. To verify this, some students set up a loudspeaker L and receiver R on an open flatcar, as in Fig. 1.; by connecting the electrical signals from L and R to an oscilloscope, they can measure the time for a sound to travel from L to R and hence find its speed u′ (relative to the car). (a) Derive an expression for u′ in terms of u, v, and θ′, where v is the car’s speed through the air and θ′ is the angle between v and LR. (We call this θ′ since it is the angle between v and u′, the velocity of the sound measured in the frame of the car.) [Hint: Draw a velocity-addition triangle to represent the relation u = u′ + v. The law of cosines should give you a quadratic equation for u′.] (b) If the students vary the angle θ′ from 0 to 180°, what are the largest and smallest values of u′? (c) If v is about 15 m/s (roughly 30 mi/h), what will be the approximate percent variation in u′? Would this be detectable?

FIGURE 1