🎡AP Physics 1
4. In Scenario 1, a boat is observed from the riverbank (an inertial reference frame). The river current flows due east at a constant speed of . The boat's speed relative to the water is constant at and is directed due north, perpendicular to the current, as shown in Figure 1. The boat starts at the south bank at point S and travels to the north bank, which is away. Neglect air resistance and assume the current speed is constant everywhere.
In Scenario 2, the same boat is again observed from the riverbank. The current is still due east at , and the boat's speed relative to the water is still . However, the boat is now aimed west of north (upstream), as shown in Figure 2. The boat again starts at point S on the south bank and travels to the north bank a distance away. Neglect air resistance and assume the current speed is constant everywhere.
Figure 1. Scenario 1: Boat aimed due north while river current flows due east.
Figure 2. Scenario 2: Boat aimed 22° west of north while river current flows due east.
Refer to Figure 2. Refer to Figure 1. Indicate whether the magnitude of the boat's velocity relative to the riverbank in Scenario 1, , is greater than, less than, or equal to the magnitude of the boat's velocity relative to the riverbank in Scenario 2, by writing one of the following in your answer booklet.
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Justify your answer by describing how the boat's velocity relative to the water and the current velocity add as vectors, including how the perpendicular components affect the magnitude. Use qualitative reasoning beyond referencing equations.
Starting with the definition of a vector sum, derive an expression for the magnitude of the boat's velocity relative to the ground, . Express your answer in terms of , , , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Consider the general case in which the current has speed due east and the boat has speed relative to the water aimed at an angle west of north.
Indicate whether the expression for you derived in part B is or is not consistent with the claim made in part A. Briefly justify your answer by referencing how depends on in your expression.
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