Lesson 1: Displacement vs. Distance: Defining Position in 1D Motion

1. What is the actual difference between distance and displacement?

Distance is a scalar quantity that measures the total ground covered during motion (the length of the path). It ignores direction. Displacement is a vector quantity that measures the straight-line change in position. It depends only on the start and end points.

Formula: Displacement Δx = x_final - x_initial

2. Can my displacement be zero even if I traveled a long distance?

Yes. Displacement is purely about where you finish relative to where you started. If you run a 400 m lap around a track and finish at the exact starting line, your distance is 400 m, but your displacement is 0 m.

3. Does a negative displacement mean I traveled a shorter distance?

No. In Physics, the positive (+) and negative (-) signs indicate direction, not value. If we define "Right" as positive, a displacement of -5 m simply means you moved 5 meters to the left. It is not "less than" zero; it is a movement in the opposite direction

Lesson 2: Average Velocity & Average Speed: Which formula to use?

4. Why can't I just average the two speeds (v₁ + v₂) / 2 to find the average velocity?

This arithmetic mean only works if you travel at those speeds for equal amounts of time. If you travel for equal distances at different speeds (e.g., driving to school and back), you spend more time driving the slower leg. You must always use the definition:

v_avg = (Total Displacement) / (Total Time)

5. Is it possible to have a high average speed but zero average velocity?

Yes. Since average velocity relies on displacement (Δx), any round trip results in zero displacement and therefore zero average velocity. However, because you covered distance, your average speed is a positive number.

6. Which formula do I use if I am given distances and speeds but no time?

You must calculate the time for each leg of the trip first.

• Step 1: Use t = d/v to find the time for each segment (t₁ and t₂).
• Step 2: Add them to get total time.
• Step 3: Divide total distance by total time
  

Lesson 3: Instantaneous Velocity: Using Limits to Find Speed at an Instant

7. How is instantaneous velocity different from average velocity?

Average velocity is the rate of position change over a distinct time interval (like an hour). Instantaneous velocity is the rate of change at a specific, single moment (like a speedometer reading). Geometrically, average velocity is the slope of the secant line, while instantaneous velocity is the slope of the tangent line.

8. In the limit formula, why does Δt have to approach zero?

To find velocity at an "instant," we need the time interval to vanish. Since we cannot divide by zero, we use a limit where Δt becomes infinitesimally small (Δt → 0). This shrinks the measurement window until it represents a single point in time.

9. Can instantaneous velocity be negative?

Yes. A negative instantaneous velocity simply means that at that specific split-second, the object is moving in the negative direction (e.g., backwards or downwards).

Lesson 4:  Acceleration on v-t Graphs: Finding Slope & Instantaneous Values

10. What does the slope of a velocity-time graph actually represent?

The slope represents acceleration (a). Since slope is "rise over run" (Δy/Δx), on this graph it becomes Δv/Δt, which is the exact definition of acceleration.

• Steep slope = High acceleration.
• Horizontal line (slope = 0) = Constant velocity.

11. If the line is straight on a v-t graph, does that mean acceleration is constant?

Yes. A straight diagonal line has a constant slope, which means the acceleration is constant. This is often called Uniformly Accelerated Motion (UAM).

12. Does a negative slope on a v-t graph always mean the object is slowing down?

No. This is the Double Negative Trap.

• If velocity is positive (+) and slope is negative (-): You are slowing down.
• If velocity is negative (-) and slope is negative (-): You are speeding up in the negative direction (moving faster away from the origin).

13. How do I find acceleration if the graph is a curve, not a straight line?

A curve means acceleration is changing. To find acceleration at a specific time, you must draw a tangent line to the curve at that point and calculate the slope of that tangent line.

Lesson 5:  The 3 Equations of Motion: Deriving Formulas for Constant Acceleration

14. How do I know which of the 3 kinematic equations to pick for a problem?

Use the "Missing Variable" Strategy. List your knowns and the one unknown you need.

1. If Time (t) is missing: Use v² = v₀² + 2aΔx
2. If Final Velocity (v) is missing: Use Δx = v₀t + ½at²
3. If Displacement (Δx) is missing: Use v = v₀ + at

15. How do you derive the equation v² = v₀² + 2aΔx if you don't have time (t)?

You combine the first two equations to eliminate 't'.

• Solve v = v₀ + at for t → t = (v - v₀)/a
• Substitute this expression for t into the displacement equation.
• Simplify the algebra to get the "time-independent" equation.

16. Can I use these equations if the acceleration is changing (like with air resistance)?

Absolutely not. The "Big 3" Kinematic equations are strictly boundary-limited to Constant Acceleration. If 'a' changes (jerk), you must use Calculus (integrals).

17. What happens to the equations if my initial velocity (v₀) is zero?

The math becomes much easier because any term with v₀ vanishes.

• v = at
• Δx = ½at²

Lesson 6: Free Fall & Gravity: Why Objects Accelerate at 9.8 m/s²

18. Is the acceleration due to gravity (g) positive 9.8 or negative 9.8?

'g' represents the magnitude of gravity, which is positive 9.8 m/s². However, in equations, we assign a sign based on direction. If "Up" is positive, gravity acts "Down," so a = -9.8 m/s².

19. If I throw a ball up, is the acceleration zero at the very top?

No. This is the #1 student misconception. At the peak, the velocity is zero for an instant, but acceleration is still 9.8 m/s² down. Gravity does not turn off just because the object stops moving momentarily.

20. Why do heavy and light objects hit the ground at the same time (assuming no air)?

This is the Equivalence Principle. A heavy object has more gravitational force pulling it (F_g), but it also has more mass (inertia) resisting that pull. The ratio F/m always simplifies to 'g'. The mass cancels out.

21. Does "free fall" still apply if the object is moving upwards?

Yes. In physics, "Free Fall" means the only force acting on the object is gravity. It does not matter if the object is moving up, down, or sideways.

Lesson 7:  Graphical Integration: Finding Displacement from v-t Graph Area

22. How do I calculate displacement from the area of a v-t graph?

The area between the graph line and the x-axis (time axis) represents the change in position (displacement).

• • Area = Height (velocity) × Width (time) = Displacement.

23. Do I subtract the area if it is below the x-axis?

It depends on the question:

• For Displacement: Yes. Treat area below the axis as negative (-). Subtract it from the positive area.
• For Total Distance: No. Treat all areas as positive magnitudes (absolute value) and add them together.

24. Does the area under an acceleration-time graph give velocity or change in velocity?

It gives the Change in Velocity (Δv). To find the actual final velocity, you must add this area to your starting velocity (v_final = v_initial + Area).

25. What if the shape isn't a perfect triangle or rectangle?

You can use the "Trapezoid Rule" or break the complex shape into simpler parts (a rectangle plus a triangle on top). Calculate the area of each shape individually and sum them up