Required information
A bicycle travels $3.40 \mathrm{~km}$ due east in $0.600 \mathrm{~h}$, then $9.10 \mathrm{~km}$ at $15.0^{\circ}$ east of north in $0.370 \mathrm{~h}$, and finally another $3.40 \mathrm{~km}$ due east in $0.600 \mathrm{~h}$ to reach its destination. The time lost in turning is negligible. Assume that east is in the $+x$-direction and north is in the $+y$-direction.
What is the direction of the average velocity for the entire trip? Enter the answer as an angle in degrees north of east.
north of east
Answer
Thus, the direction of the average velocity for the entire trip is approximately $8.59^{\circ}$ north of east. So, the final answer is \(\boxed{8.59^{\circ}}\).
Step 1 ：Given that a bicycle travels $3.40 \mathrm{~km}$ due east in $0.600 \mathrm{~h}$, then $9.10 \mathrm{~km}$ at $15.0^{\circ}$ east of north in $0.370 \mathrm{~h}$, and finally another $3.40 \mathrm{~km}$ due east in $0.600 \mathrm{~h}$ to reach its destination. The time lost in turning is negligible. Assume that east is in the $+x$-direction and north is in the $+y$-direction.
Step 2 ：The average velocity is the total displacement divided by the total time. The displacement is a vector quantity, so we need to calculate the x and y components of the displacement for each leg of the trip, and then add them up to get the total displacement.
Step 3 ：The x and y components of the displacement for the first leg of the trip are $x_1 = 3.4$ and $y_1 = 0$ respectively.
Step 4 ：For the second leg of the trip, the x and y components of the displacement are $x_2 = 8.789925019230521$ and $y_2 = 2.3552533104329387$ respectively.
Step 5 ：The x and y components of the displacement for the third leg of the trip are $x_3 = 3.4$ and $y_3 = 0$ respectively.
Step 6 ：The total x and y displacements are $x_{total} = 15.589925019230522$ and $y_{total} = 2.3552533104329387$ respectively.
Step 7 ：The direction of the average velocity is the arctangent of the ratio of the total y displacement to the total x displacement, which is $8.591013066841251$ degrees.
Step 8 ：Thus, the direction of the average velocity for the entire trip is approximately $8.59^{\circ}$ north of east. So, the final answer is \(\boxed{8.59^{\circ}}\).
