When a bat uses echolocation to determine the distance to an insect, it sends out a sound wave and waits to see how long the sound takes to echo back. Suppose now that a bat hears its echo 0.06 LaTeX: ss after it emitted the sound. If the speed of sound is 344.9 LaTeX: \frac{m}{s}m s, how far away is the insect in meters?

Space-time metric ds2= -dt2+at2dx2+bt2dy2+ct2dz2 (a)how does the area of a rectangle with vertices at ( x=0, y=0, z=0 ), (1,0,0 ), ( 0,1,0 ) and (1,1,0 ) changes with time? (b) if a photon moving in the x direction starts from the origin at t=1, when does it reach x=l? (ds=0 for the trajectory of a photon.) (c) the proper time for a particle is defined by cd=ds. what is the proper time for a particle to move along the trajectory ( x(t)=t, y(t)=0, z(t)=0 ) from t=0 to 1 ? what is the condition on the parameter a so that the velocity of the particle never exceeds the speed of light ?

Standing side by side ,you and a friend step off a bridge at different times and fall for 1.6s to the water below. your friend goes first, and you follow after he has dropped a distance of 2.0m. when your friend hits the water, is the seperation between the two of you 2.0m or more than 2.0m? verify your answer with a calculation.

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