Relative velocity in two dimensions Transitional Reasoning Download PDF

Questions

1. 1.

   A mobile meteorological laboratory drives along a flat, straight road due east at a constant speed of 15 m/s relative to the ground. A weather balloon released nearby ascends vertically at a steady speed of 8.0 m/s while simultaneously being carried horizontally due east by wind at 21 m/s, both measured relative to the ground. What is the velocity of the balloon as measured by an optical tracking sensor mounted on the moving laboratory? (Take $\sin 37^\circ = \cos 53^\circ = 0.60$ and $\sin 53^\circ = \cos 37^\circ = 0.80$.)

   • A) 10 m/s at an angle of $37^\circ$ above the horizontal directed east
   • B) 10 m/s at an angle of $53^\circ$ above the horizontal directed east
   • C) 10 m/s at an angle of $53^\circ$ above the horizontal directed west
   • D) 10 m/s at an angle of $37^\circ$ above the horizontal directed west
2. 2.

   On the horizontal floor of an automated distribution center, two mobile robots navigate along programmed pathways. Robot 1 moves with a constant speed of 2.0 m/s directed 60° north of east relative to the floor. An optical tracking sensor on Robot 1 detects Robot 2 moving with a constant velocity of 2.0 m/s directed 60° south of east relative to Robot 1. What is the velocity of Robot 2 relative to the floor? (Take cos 60° = 0.50 and sin 60° = 0.87.)

   • A) 3.5 m/s directed due south
   • B) 4.0 m/s directed due east
   • C) 2.0 m/s directed due east
   • D) 2.8 m/s directed 45° south of east
3. 3.

   A passenger train travels due east along a straight track at a constant speed of 18 m/s relative to the ground, while a drone is observed from the train to be moving at 16 m/s directed due north relative to the train. On a parallel road, a security vehicle drives due east at a constant speed of 30 m/s relative to the ground. What is the velocity of the drone as measured by an observer in the security vehicle? (Take $\sin 53^\circ = \cos 37^\circ = 0.80$ and $\cos 53^\circ = \sin 37^\circ = 0.60$.)

   • A) 20 m/s directed $37^\circ$ north of west
   • B) 20 m/s directed $53^\circ$ north of east
   • C) 20 m/s directed $53^\circ$ north of west
   • D) 25 m/s directed $53^\circ$ north of west
4. 4.

   A naval helicopter takes off from the flight deck of a support ship that cruises due east at a constant speed of 24 m/s relative to the shore. Once airborne, the helicopter maintains a constant velocity of 20 m/s directed $53^\circ$ north of west relative to the moving ship. What is the velocity of the helicopter as measured by a stationary radar station on the shore? (Take $\cos 53^\circ = 0.60$ and $\sin 53^\circ = 0.80$.)

   • A) 44 m/s directed $53^\circ$ north of east
   • B) 20 m/s directed $37^\circ$ north of east
   • C) 20 m/s directed $53^\circ$ north of east
   • D) 20 m/s directed $53^\circ$ north of west
5. 5.

   On a horizontal surveillance grid over a solar power facility, east is defined as the $+x$-direction and north as the $+y$-direction. Patrol drone Alpha cruises at a constant speed of 10 m/s at an angle of $53^\circ$ north of east, while patrol drone Bravo cruises at a constant speed of 10 m/s at an angle of $53^\circ$ north of west. What is the velocity of drone Bravo relative to drone Alpha? (Take $\cos 53^\circ = 0.60$ and $\sin 53^\circ = 0.80$.)

   • A) 12 m/s directed due west
   • B) 12 m/s directed due east
   • C) 16 m/s directed due south
   • D) 20 m/s directed due west
6. 6.

   During a coastal monitoring operation, a patrol cutter (P) and an aerial drone (D) are tracked on a coordinate grid where the $+x$-axis points due east and the $+y$-axis points due north. The cutter moves across the water at a constant velocity of 10.0 m/s directed $37^\circ$ north of east, while the drone maintains level flight at a constant velocity of 21.0 m/s directed due north. What is the velocity vector of the drone relative to the cutter ($\vec{v}_{D/P}$), and what is its magnitude? (Take $\cos 37^\circ = 0.80$ and $\sin 37^\circ = 0.60$.)

   • A) $\vec{v}_{D/P} = (-8.0\hat{i} + 15.0\hat{j})\text{ m/s}$ with a magnitude of 17.0 m/s
   • B) $\vec{v}_{D/P} = (8.0\hat{i} + 27.0\hat{j})\text{ m/s}$ with a magnitude of 28.2 m/s
   • C) $\vec{v}_{D/P} = (-8.0\hat{i} + 27.0\hat{j})\text{ m/s}$ with a magnitude of 28.2 m/s
   • D) $\vec{v}_{D/P} = (8.0\hat{i} + 15.0\hat{j})\text{ m/s}$ with a magnitude of 17.0 m/s
7. 7.

   An oceanographic research vessel navigates due west at a constant speed of 12 m/s relative to the calm water. An instrument on the vessel's mast measures an apparent wind velocity of 16 m/s directed due south relative to the vessel. What are the speed and direction of the true wind velocity relative to the water? (Take $\sin 37^\circ = \cos 53^\circ = 0.6$ and $\sin 53^\circ = \cos 37^\circ = 0.8$.)

   • A) 20 m/s, directed toward 53° south of east
   • B) 28 m/s, directed toward south-west
   • C) 20 m/s, directed toward 53° south of west
   • D) 20 m/s, directed toward 37° south of west
8. 8.

   A delivery courier rides a bicycle along a straight, level road heading due east at a constant speed of 9.0 m/s. In the surrounding calm air, rain falls purely vertically downward at a terminal speed of 12.0 m/s relative to the ground. To block the falling rain as effectively as possible, the courier aligns an umbrella parallel to the apparent path of the raindrops. What is the speed of the raindrops relative to the courier, and in which direction should the umbrella be tilted from the vertical?

   • A) 7.9 m/s, tilted 37° toward the west from the vertical
   • B) 21.0 m/s, tilted 53° toward the east from the vertical
   • C) 15.0 m/s, tilted 37° toward the west from the vertical
   • D) 15.0 m/s, tilted 37° toward the east from the vertical
9. 9.

   A ferry needs to cross a straight 240 m wide river to dock at a landing pier directly opposite its launch point. The river current moves uniformly downstream at a constant speed of 1.5 m/s, and the ferry cruises at a steady speed of 2.5 m/s relative to the water. In order to follow a straight path perpendicular to the riverbanks, what must be the magnitude of the ferry's resultant velocity relative to the ground, and how long does the crossing take?

   • A) 2.0 m/s and 120 s
   • B) 2.0 m/s and 96 s
   • C) 2.5 m/s and 96 s
   • D) 2.9 m/s and 83 s
10. 10.

    A harbor monitoring station tracks two autonomous surface vessels moving across calm water. Vessel P maintains a constant velocity of 15 m/s directed due north, while Vessel Q travels at a constant velocity of 20 m/s directed due east. What is the velocity of Vessel Q as measured by an observer aboard Vessel P?

    • A) 25 m/s at an angle of 37° south of east
    • B) 25 m/s at an angle of 37° north of east
    • C) 25 m/s at an angle of 37° north of west
    • D) 5 m/s directed due east