Motion in a Plane: Putting It All Together
Test how well you can connect vector operations, projectile motion, circular motion, and relative velocity. These medium-difficulty questions emphasize choosing a useful approach and interpreting motion through components.
Q1. Which quantity is a scalar rather than a vector?
Speed has magnitude but no direction, whereas displacement, velocity, and acceleration are vectors.
Q2. A vector of magnitude ##v## makes an angle ##\theta## above the horizontal. Which expression gives its horizontal component?
For an angle measured from the horizontal, the horizontal component is the adjacent component: ##v_x=v\cos\theta##.
Q3. A ball is launched at ##25\,\text{m/s}## at ##40^\circ## above the horizontal. Approximately what is its initial horizontal velocity, using ##\cos 40^\circ\approx0.766##?
Resolve the launch velocity horizontally: ##v_x=25\cos40^\circ\approx19.2\,\text{m/s}##.
Q4. In the absence of air resistance, what happens to a projectile's horizontal velocity during its flight?
Gravity acts vertically, so the horizontal velocity remains constant when air resistance is neglected.
Q5. At the highest point of a projectile's path, which statement is correct?
At the top of the trajectory, the vertical component of velocity is momentarily zero. The horizontal component remains constant in the ideal model.
Q6. Why can projectile motion be analyzed as two simpler one-dimensional motions?
Resolving motion into perpendicular components lets the horizontal and vertical motions be analyzed separately; gravity affects the vertical component.
Q7. Two vectors are added using the triangle method. Which statement about their sum is also true under the parallelogram method?
Vector addition is commutative: ##\vec{A}+\vec{B}=\vec{B}+\vec{A}##. The triangle and parallelogram constructions represent the same resultant.
Q8. How should the subtraction ##\vec{A}-\vec{B}## be interpreted when solving a vector problem?
Vector subtraction is addition of the opposite vector: ##\vec{A}-\vec{B}=\vec{A}+(-\vec{B})##.
Q9. A ferry moves relative to the water, and the water moves relative to the shore. Which vector relationship gives the ferry's velocity relative to the shore?
Relative velocities combine as vectors: ferry relative to shore equals ferry relative to water plus water relative to shore.
Q10. A drone moves in a circle at constant speed. Which statement best describes its velocity?
Velocity includes direction as well as speed. In circular motion, the direction changes continuously, so velocity changes even when speed is constant.
Q11. For uniform circular motion, in what direction does the acceleration point?
In uniform circular motion, the acceleration is directed inward, toward the center of the circular path.
Q12. A satellite moves in a circular orbit. Which feature of the motion continually changes the direction of its velocity?
The inward acceleration continually changes the direction of the satellite's velocity, keeping its path curved.
Q13. A student adds the magnitudes of two perpendicular vectors and calls that number the resultant magnitude. What is the main error?
For perpendicular vectors, the resultant magnitude follows the Pythagorean theorem; simply adding their magnitudes gives the wrong result.
Q14. A ball is launched at ##25\,\text{m/s}## at ##40^\circ## above the horizontal. Approximately what is its initial vertical velocity, using ##\sin40^\circ\approx0.643##?
The vertical component is ##v_y=25\sin40^\circ\approx16.1\,\text{m/s}##.
Q15. A ball is struck toward a boundary, and air resistance is ignored. Which pair of ideas is most directly useful for describing its flight?
Resolving the launch velocity separates the projectile's horizontal and vertical motions, with gravity affecting the vertical component.
Q16. A current flows east while a ferry moves north relative to the water. What is the most useful way to determine the ferry's direction of travel relative to shore?
The ferry's velocity relative to shore is the vector sum of its velocity relative to the water and the water's velocity relative to shore. The components determine its direction.
Q17. A velocity vector has components ##v_x=3\,\text{m/s}## and ##v_y=4\,\text{m/s}##. What is its magnitude?
Use the Pythagorean theorem to recombine perpendicular components: ##v=\sqrt{3^2+4^2}=5\,\text{m/s}##.
Q18. Which sequence is a sound general strategy for a two-dimensional motion problem?
Separating vector quantities into components gives a systematic route: solve the independent component problems and recombine the results when required.
Q19. A ball moves around a circular path at constant speed. Which statement distinguishes speed from velocity in this situation?
Speed is the magnitude of velocity and can remain constant, while velocity changes because its direction changes along the circle.
Q20. Which statement best connects vector resolution with the wider study of motion in a plane?
Component resolution is a shared tool: it helps organize vector quantities in projectile, circular, and relative-motion problems.