Vectors and Motion in Two Dimensions

Learning Goals

After this lesson, you should be able to:

• resolve a vector into signed perpendicular components;
• reconstruct vector magnitude and direction from components;
• add vectors component by component;
• model ideal projectile motion with independent horizontal and vertical equations;
• connect trajectory shape with component velocity and acceleration graphs;
• solve horizontal- and angled-launch problems with a shared time; and
• design an experiment that tests a projectile-motion prediction.

1. One Vector, Two Components

If angle \(\theta\) is measured counterclockwise from the positive \(x\)-axis, a vector of magnitude \(A\) has components

The components are signed scalars along the chosen axes. They are not additional vectors that must be added to the original; rather, \(\vec A=A_x\hat i+A_y\hat j\). Angles measured from another axis require a different sine–cosine assignment or a careful triangle.

2. Reconstructing Magnitude and Direction

Use the signs of \(A_x\) and \(A_y\) to choose the correct quadrant. A calculator's simple inverse tangent may return only a reference angle. A sketch or a quadrant-aware calculation prevents a direction error.

Example: \(A_x=-6\) and \(A_y=+8\) give \(A=10\). The vector lies in quadrant II, so its direction is about \(127^\circ\) from \(+x\), not \(-53^\circ\).

3. Add Vectors by Components

Resolve every vector using the same axes, add like components, then reconstruct the resultant:

Never add one vector's magnitude directly to another vector's \(x\)-component. The quantities must refer to the same axis before they can be combined algebraically.

5. The Ideal Projectile Model

After release, model the projectile as an object acted on only by uniform gravity. Ignore air resistance and take upward as positive:

The two component motions are independent but share the same time \(t\). Gravity changes vertical velocity; it does not gradually “use up” horizontal velocity in this model.

6. Component Graphs Tell One Story

A top-view horizontal graph and a side-view vertical graph are not separate events. Match points using the same timestamps.

7. Worked Example: Horizontal Launch

Question: A ball rolls horizontally from a \(20\,\mathrm m\) high platform at \(8.0\,\mathrm{m\,s^{-1}}\). Ignore air resistance and use \(g=10\,\mathrm{m\,s^{-2}}\). Find flight time, horizontal range, and impact velocity.

Vertical motion determines time:

Horizontal motion determines range:

Recombine impact components:

Answer: The ball is airborne for \(2.0\,\mathrm s\), lands \(16\,\mathrm m\) horizontally from the launch point, and impacts at about \(21.5\,\mathrm{m\,s^{-1}}\), directed approximately \(68^\circ\) below horizontal.

8. Worked Example: Angled Launch

A projectile is launched from level ground at \(20\,\mathrm{m\,s^{-1}}\), \(30^\circ\) above horizontal. It lands at its launch height. Use \(g=10\,\mathrm{m\,s^{-2}}\).

At the highest point, \(v_y=0\), so \(t_{up}=v_{0y}/g=1.0\,\mathrm s\). Symmetry applies because launch and landing heights match, giving total time \(2.0\,\mathrm s\).

Answer: The projectile rises \(5.0\,\mathrm m\) above launch level and travels about \(34.6\,\mathrm m\) horizontally.

9. Functional Dependence: Horizontal Range

For a horizontal launch from height \(h\), vertical motion gives \(t=\sqrt{2h/g}\). Substitution into horizontal motion gives

Therefore, doubling \(v_{0x}\) doubles \(R\). Quadrupling \(h\) doubles both flight time and range. Increasing horizontal speed does not change flight time when height and vertical initial velocity remain the same.

10. Vector and Projectile Problem Routine

1. Draw axes and define positive directions.
2. Resolve every needed vector before using kinematics.
3. Create separate columns for \(x\) and \(y\).
4. Record \(a_x\), \(a_y\), initial components, and displacement components.
5. Use the component with enough information to find the shared time.
6. Carry that time into the other component equation.
7. Recombine components only when the requested answer is a magnitude and direction.
8. Check quadrant, units, graph shape, and model assumptions.

11. AP Reasoning: Dropped Versus Launched

Two identical balls start at the same height at the same instant. One is dropped; the other is launched horizontally. Ignore air resistance.

Claim: they reach the floor simultaneously. Evidence: both have the same \(y_0\), \(v_{0y}=0\), and \(a_y=-g\). Reasoning: horizontal velocity changes only the horizontal position; the vertical equation determining flight time is identical for both balls.

12. Investigation: Predict the Landing Point

Launch a ball horizontally from a table using a repeatable low-energy launcher. Measure table height and horizontal launch speed with video or two position markers.

1. Predict flight time from vertical data only.
2. Predict horizontal range using the measured \(v_{0x}\).
3. Mark the predicted point and an uncertainty interval before launching.
4. Perform repeated trials and record landing positions.
5. Compare the mean landing position with the prediction.
6. Discuss how speed variation, height measurement, camera perspective, launch angle, and air resistance affect the result.

13. Common Traps and Fast Checks

• Trap: using sine for \(x\) automatically. Check: identify the side adjacent to the stated angle.
• Trap: discarding component signs before recombination. Check: signs determine the quadrant.
• Trap: using different times for horizontal and vertical motion. Check: both components describe the same object during the same flight.
• Trap: setting acceleration to zero at the highest point. Check: only \(v_y\) is zero there; \(a_y=-g\).
• Trap: assuming \(v_x\) decreases because the path curves. Check: with no air resistance, \(a_x=0\).
• Trap: doubling total speed and claiming range always doubles. Check: identify which velocity component and which other variables remain fixed.

14. Key Takeaways

• Perpendicular components add to reproduce the original vector.
• Resolve, calculate by axis, then recombine.
• Ideal projectile motion has constant horizontal velocity and constant downward acceleration.
• Horizontal and vertical components evolve independently but share time.
• The highest point has \(v_y=0\), not zero total velocity or zero acceleration.
• Signs, quadrants, units, and limiting behavior are essential checks.