Graphical Analysis of Motion in a Plane Quiz
Test your understanding of how component-wise position-time and velocity-time graphs reveal motion in two dimensions. Interpret graph shapes, slopes, and areas, and apply them to projectile motion.
Q1. What physical quantity is represented by the slope of an ##x(t)## position-time graph?
The slope of position versus time gives velocity. For an ##x(t)## graph, it gives the x-component of velocity.
Q2. What does the slope of a ##v_y(t)## graph represent?
The slope of a velocity-time graph gives the acceleration component corresponding to that velocity. Thus the slope of ##v_y(t)## gives the y-component of acceleration.
Q3. A projectile moves without air resistance. What shape does its horizontal position-time graph have?
With no air resistance, horizontal velocity is constant, so horizontal position changes linearly with time. Its position-time graph is a straight line.
Q4. For a projectile under constant downward acceleration, what is the shape of its vertical position-time graph when upward is positive?
Constant downward acceleration makes vertical position vary quadratically with time. With upward chosen as positive, the position-time curve is a downward-opening parabola.
Q5. A particle's x-position changes from ##4\ \mathrm{m}## to ##16\ \mathrm{m}## in ##3\ \mathrm{s}## along a straight position-time graph. What is its x-velocity?
The slope is the change in x-position divided by elapsed time: ##(16-4)/3=4\ \mathrm{m/s}##. A straight graph indicates this velocity is constant.
Q6. The y-velocity changes linearly from ##12\ \mathrm{m/s}## to ##-8\ \mathrm{m/s}## over ##4\ \mathrm{s}##. What is the y-acceleration?
The slope of the velocity-time graph is ##(-8-12)/4=-5\ \mathrm{m/s^2}##, which is the y-acceleration.
Q7. A horizontal velocity-time graph stays at ##6\ \mathrm{m/s}## for ##5\ \mathrm{s}##. What is the x-displacement during that interval?
The area under a velocity-time graph gives displacement. Here the area is ##6\times5=30\ \mathrm{m}##.
Q8. The y-velocity increases linearly from zero to ##10\ \mathrm{m/s}## over ##2\ \mathrm{s}##. What is the y-displacement over this interval, given that the graph remains above the time axis?
The displacement equals the area under the velocity-time graph. The area is a triangle: ##\dfrac{1}{2}\times2\times10=10\ \mathrm{m}##.
Q9. Which statement correctly describes a curved position-time graph for one component of motion?
A changing slope on a position-time graph means the component's velocity changes. The curve alone does not determine the other component's motion.
Q10. A straight, sloping ##v_x(t)## graph indicates that the x-component of acceleration is:
The slope of ##v_x(t)## is x-acceleration. A straight velocity-time graph has a constant slope, so the x-acceleration is constant.
Q11. At the highest point of a projectile's path, which statement about its vertical motion is correct?
At the top of the projectile's path, the vertical velocity is momentarily zero. The vertical position is at its maximum, while downward acceleration continues.
Q12. A projectile's horizontal velocity is constant and its vertical velocity changes steadily. Which pair of position-time graphs is consistent with this motion?
Constant horizontal velocity produces a straight-line x-position graph. Steadily changing vertical velocity corresponds to constant vertical acceleration and a parabolic y-position graph.
Q13. A learner sees a downward-sloping ##x(t)## graph and concludes that the particle's x-position is negative. What is the mistake?
The slope of ##x(t)## gives x-velocity. A negative slope means x-position is decreasing with time; it does not by itself say whether x-position is positive or negative.
Q14. A component's position-time graph is a straight line with slope ##-3\ \mathrm{m/s}##. Which equation could describe that component's position?
A straight position-time graph with slope ##-3\ \mathrm{m/s}## has the form ##x(t)=x_0-3t##, where ##x_0## is the initial position.
Q15. A velocity-time graph has a positive slope throughout an interval. What does this establish about the corresponding acceleration component?
The slope of a velocity-time graph is the corresponding acceleration. A positive slope therefore means positive acceleration.
Q16. A projectile's ##v_y(t)## graph crosses the time axis. What does the crossing indicate?
Where ##v_y(t)## crosses zero, the vertical velocity is zero at that instant and changes sign. For a projectile rising and then falling, this occurs at the peak.
Q17. Why are the x- and y-components often graphed separately when analysing motion in a plane?
Horizontal and vertical motion can be analysed independently. Their component-wise graphs can then be used together to describe the complete motion.
Q18. A flat ##y(t)## graph over a time interval indicates that the particle's y-position is:
A horizontal position-time graph has zero slope, so the y-position does not change during that interval and the y-velocity is zero.
Q19. A ball travels horizontally off a table with air resistance ignored. Which graph description best matches its motion after leaving the table?
Ignoring air resistance, horizontal velocity stays constant, so x-position is linear in time. Gravity accelerates the ball vertically, so y-position follows a parabola.
Q20. A velocity-time graph for one component lies partly above and partly below the time axis. What must be considered when finding displacement from its area?
Displacement is the signed area under the velocity-time graph. Areas below the time axis count as negative, so they must be combined with the positive areas.