Test your understanding of two-dimensional kinematics. These questions explore how constant-acceleration motion can be analyzed independently along perpendicular axes and then combined using their shared time interval.
Q1. When acceleration is constant, how can motion in two perpendicular directions be analyzed?
With constant acceleration, the equations can be resolved into separate scalar equations for each perpendicular component.
Q2. A particle has position vector ##\vec{r}=x\hat{i}+y\hat{j}##. What do ##x## and ##y## represent?
The coefficients of ##\hat{i}## and ##\hat{j}## specify the position components along the two chosen axes.
Q3. In the vector expression ##\vec{a}=a_x\hat{i}+a_y\hat{j}##, what do ##a_x## and ##a_y## describe?
The coefficients of the unit vectors give the acceleration components along their respective axes.
Q4. Why does vertical motion not change the horizontal coordinate of a particle moving along perpendicular axes?
Orthogonal components do not contribute to each other's projections, so each coordinate is determined by its own component equations.
Q5. For a constant-acceleration problem, which quantity provides the shared link between the component equations?
The horizontal and vertical equations describe the same motion over the same time interval, so time links the component solutions.
Q6. A particle has horizontal velocity ##u_x## and constant horizontal acceleration ##a_x##. Which equation gives its horizontal velocity after time ##t##?
The velocity equation for a constant acceleration component is ##v_x=u_x+a_xt##.
Q7. A particle's initial vertical velocity is ##u_y## and its vertical acceleration is ##a_y##. Which expression gives its vertical displacement after time ##t##?
Applying the displacement equation to the vertical component gives ##\Delta y=u_yt+\dfrac{1}{2}a_yt^2##.
Q8. If a particle's acceleration is constant, how should the acceleration components be treated during a given time interval?
A constant acceleration vector has constant components along the chosen fixed axes.
Q9. A particle starts with horizontal velocity ##3\ \mathrm{m\,s^{-1}}## and has zero horizontal acceleration. What is its horizontal velocity after ##4## seconds?
Using ##v_x=u_x+a_xt## with ##u_x=3\ \mathrm{m\,s^{-1}}## and ##a_x=0## gives ##v_x=3\ \mathrm{m\,s^{-1}}##.
Q10. A particle's vertical velocity changes from ##2\ \mathrm{m\,s^{-1}}## to ##10\ \mathrm{m\,s^{-1}}## under constant vertical acceleration ##4\ \mathrm{m\,s^{-2}}##. How long does the change take?
From ##v_y=u_y+a_yt##, ##t=\dfrac{v_y-u_y}{a_y}=\dfrac{10-2}{4}=2\ \mathrm{s}##.
Q11. A particle has ##u_x=2\ \mathrm{m\,s^{-1}}##, ##a_x=1\ \mathrm{m\,s^{-2}}##, and moves for ##3\ \mathrm{s}##. What is its horizontal displacement?
Use ##\Delta x=u_xt+\dfrac{1}{2}a_xt^2##. Substitution gives ##\Delta x=2(3)+\dfrac{1}{2}(1)(3^2)=10.5\ \mathrm{m}##.
Q12. A particle's horizontal and vertical equations produce different displacement values over the same interval. What does this indicate?
The two displacements are different coordinate components. They need not have equal values, even though they refer to the same interval.
Q13. A particle's position changes by ##\Delta x## horizontally and ##\Delta y## vertically. How is its displacement vector represented?
The displacement vector is formed by combining the horizontal and vertical components with their corresponding unit vectors.
Q14. A curved path is produced when the horizontal and vertical position components change simultaneously. What does the path represent?
The curved trajectory is the combined result of the independently evolving horizontal and vertical position components.
Q15. A ball moves horizontally with zero horizontal acceleration while a vertical acceleration acts on it. What happens to its horizontal velocity in this model?
With ##a_x=0##, the horizontal equation ##v_x=u_x+a_xt## gives ##v_x=u_x## throughout the interval.
Q16. Which procedure is the most suitable first step for solving a constant-acceleration problem in two dimensions?
Separating the equations by axis organizes the known quantities and unknowns while preserving the independence of the components.
Q17. A particle has initial velocity components ##u_x## and ##u_y##, and constant acceleration components ##a_x## and ##a_y##. Which expression gives its velocity vector after time ##t##?
Apply ##v_x=u_x+a_xt## and ##v_y=u_y+a_yt## separately, then combine the components with ##\hat{i}## and ##\hat{j}##.
Q18. An object's initial vertical position and vertical velocity, constant vertical acceleration, and a later vertical position are known. If the vertical displacement equation has a unique physically admissible nonnegative solution, which quantity can be determined?
The vertical displacement equation relates the known vertical displacement to initial vertical velocity, constant vertical acceleration, and elapsed time. Under the stated uniqueness condition, it determines the elapsed time. The other options are not quantities determined by this equation.
Q19. In a graphical representation of two-dimensional motion, what does plotting horizontal position against vertical position show?
Plotting the position components together shows the trajectory traced by their combined motion.
Q20. Why can equations for the two axes be solved separately and then recombined?
Vector equations split into scalar component equations along perpendicular axes. Solving those equations separately and combining their results reconstructs the motion.