Scalars & Vectors (College Board AP® Physics 1: Algebra-Based): Exam Questions

A hiker follows a path consisting of three segments:

• due east

• due north

• due west

The hiker maintains a constant average speed of while walking.

Estimate whether the hiker's average velocity is greater than, less than, or equal to their average speed.

Justify your estimate using qualitative reasoning.

Derive an equation for both the hiker’s average speed and the magnitude of their average velocity.

Does the equation derived in part b) support your qualitative reasoning from part a)? Justify qualitatively why or why not.

A skier moves down an inclined plane with an angle relative to the horizontal. The skier's initial velocity at the top of the incline is , and air resistance is negligible.

i) Draw a labeled diagram showing the skier’s velocity vector at the start and its components relative to the incline.

ii) Derive expressions for the initial velocity components parallel and perpendicular to the incline.

Two students set up an investigation measuring how distance from the pivot point affects torque. The students use a hinged door and a meter ruler in their investigation. Student A suggests that since torque is measured in newton-meters, torque is equivalent to the work done by the force on the door.

State whether you agree with Student A and justify your answer using qualitative reasoning.

A transatlantic airplane cruises at a constant velocity of heading from New York to London. Some time later, a tail wind develops and the cruising velocity increases to .

Draw the relative cruising velocities as vectors indicating the scalar multiplication factor.

Figure 1

A boat moves due north relative to the water, while the river current flows due east.

On the diagram in Figure 2, draw and label arrows that represent the velocity vectors of the boat and the river current relative to an observer on the riverbank. Clearly indicate the boat’s velocity relative to the water, the river’s velocity, and the boat’s resultant velocity.

Figure 2

i) Starting with the Pythagorean theorem, derive an expression for the magnitude of the boat’s velocity relative to the riverbank and determine its magnitude.

ii) Determine the direction of the boat’s velocity relative to the riverbank, measured counterclockwise from due east.

i) In a second trip, the boat crosses the river further downstream where the velocity of the river is . The distance between the north and south banks are equal at both locations and the boat's speed relative to the water remains the same.

Indicate whether the boat’s crossing time in this case would be greater than, less than, or equal to the time taken in the original scenario.

ii) Justify your answer by considering how the river current affects the motion in each case. Use both a mathematical argument and a conceptual explanation.