You drive 30 miles due east in a half hour. Then, you turn left and drive 30 miles north in 1 hour. What are your average speed and velocity, and what are the rectangular and polar coordinates of your position?
Accepted Solution
A:
Original position: P1=(0,0) You drive 30 miles due east in a half hour: x=+30 miles, t1=1/2 hour=0.5 hours Then, you turn left and drive 30 miles north in 1 hour: y=+30 miles, t2=1 hour Rectangular coordinates of final position: P2=(x,y)→P2=(30,30) Total time: t=t1+t2=0.5 hours+1 hour→t=1.5 hours
Average speed: S ave=d/t Total distance: d=x+y=30 miles+30 miles→d=60 miles S ave = 60 miles / (1.5 hours) S ave = 40 miles/hour
Velocity is a vector, the magnitude of this vector is the magnitude of the vector of change of position dividing by the total time t The vector of change of position: s=P1-P2=(30,30)-(0,0)=(30-0,30-0)→ s=(30,30) Magnitude of vector s=sqrt[30^2+30^2]=sqrt[30^2*2]=sqrt[30^2]*sqrt(2) Magnitude of vector s=30*sqrt(2) miles
Magnitude of velocity vector = Magnitud of vector s / t Magnitude of velocity vector = [30*sqrt(2) miles] / (1.5 hours) Magnitude of velocity vector = 20*sqrt(2) miles / hour Magnitude of velocity vector=20*1.4142 miles / hour Magnitude of velocity vector=28.284 miles/hour
Polar coordinates of your position=(r, theta) r=Magnitude of vector s=30*sqrt(2) miles theta=tan^(-1) (y/x) = tan^(-1) [(30 miles) / (30 miles)] theta=tan^(-1) (1)→theta=45°=Pi/4 (Pi=3.1416) Polar coordinates of your position: ( 30*sqrt(2) miles, 45°) Polar coordinate of your position: ( 30*sqrt(2) miles, Pi/4 )
Answers: Average speed: 40 miles / hour Velocity: 20*sqrt(2) miles / hour = 28.284 miles / hour Rectangular coordinates of your position = (30,30) Polar coordinates of your position=(30*sqrt(2) miles,45°) Polar coordinates of your position=(30*sqrt(2) miles,Pi/4)