Lesson 1:  Position & Displacement Vectors: Tracking Motion in a 2D Plane

1. What is the difference between a position vector and a displacement vector in 2D?

Think of the position vector as your current GPS coordinates—it points from the Origin (0,0) to where you are right now. Displacement is the arrow drawn from where you started to where you ended. If you run a full circle and return to the start, your final position vector is the same as your initial one, but your total displacement is zero.

2. How do I calculate the total distance traveled vs. the magnitude of displacement?

These are totally different! Distance is the reading on your odometer—it counts every step you took along the path (the arc length). Displacement magnitude is the straight-line distance (the "as the crow flies" distance) calculated using the Pythagorean theorem on your change in x and change in y: sqrt(delta_x² + delta_y²).

3. Why do we use unit vectors i and j? Can't we just use x and y?

We use i and j to do math with directions safely. "x" and "y" are just numbers (scalars), but i and j act like labels that say "this number belongs to the horizontal direction" and "this number belongs to the vertical direction." They prevent you from accidentally adding horizontal apples to vertical oranges.

4. If I walk North then East, how do I find the direction of my displacement?

You use trigonometry. Specifically, theta = tan⁻¹(y / x). However, be careful: your calculator assumes you are in Quadrant 1 or 4. If you walked West (negative x), you must manually add 180° to the calculator's result to get the correct angle from the positive x-axis.

5. Can a displacement vector be negative?

A vector itself isn't "negative" or "positive"—it has a magnitude (always positive) and a direction. However, its components can be negative. If the x-component is -5m, it just means the displacement points 5 meters to the left.

Lesson 2: Velocity in 2D: Calculating Speed and Direction Simultaneously

6. How do I break a diagonal velocity vector into x and y components? 

This is the most common first step in physics problems. If the angle theta is measured from the horizontal (x-axis):

• Velocity in x (vx) = v * cos(theta)
• Velocity in y (vy) = v * sin(theta)

Tip: Remember "Cos" is for "Close" to the angle.

7. A boat is crossing a river with a current. How do I find its actual speed? 

This is a vector addition problem. You have the boat's velocity vector (engine speed) and the river's velocity vector (current). You add the x-components together and the y-components together. The actual speed is the hypotenuse of the resulting triangle: sqrt(total_vx² + total_vy²).

8. How do I calculate the instantaneous velocity if I have equations for x(t) and y(t)? 

You cannot just plug time into the distance formula. You must take the derivative of the x-equation to get vx and the derivative of the y-equation to get vy. Then combine them using the Pythagorean theorem to get the total speed.

9. What is the difference between average speed and average velocity in 2D? 

Average speed is the total distance traveled divided by time (a scalar number). Average velocity is the displacement vector divided by time. If you run around a track and finish where you started, your average velocity is zero, but your average speed is definitely not!

10. If an object moves in a curve, is the velocity vector ever straight? 

The velocity vector is always tangent to the path. Even if the path is curvy, at any single frozen micro-second, the object is moving in a straight line direction tangent to that curve

Lesson 3: Average Acceleration & Instantaneous Acceleration

11. What exactly is "instantaneous" acceleration compared to "average" acceleration in 2D?

Average acceleration looks at the start and end points: (final velocity - initial velocity) / time. Instantaneous acceleration is what is happening right now—it is the rate at which your velocity is changing at a specific instant. In calculus terms, it is the second derivative of your position function.

12. Is it possible for a particle to have zero velocity but non-zero acceleration in 2D?

Yes! Think of a ball thrown straight up. At the very peak of its flight, its velocity is momentarily zero. However, gravity is still pulling it down, so its acceleration is still 9.8 m/s² downwards. If acceleration were zero, the ball would hover there forever!

13. If acceleration is negative (like -9.8 m/s²), does that mean the object is slowing down?

Not necessarily. Negative acceleration just means the acceleration points in the negative direction (usually left or down). If you are falling down (negative velocity) and gravity pulls down (negative acceleration), you speed up. You only slow down if acceleration and velocity point in opposite directions.

14. Why is the acceleration vector always perpendicular to velocity in Uniform Circular Motion?

Acceleration does two things: it changes speed OR it changes direction. Parallel acceleration changes speed. Perpendicular acceleration changes direction. In uniform circular motion, the speed is constant, so there is zero parallel acceleration—all of it is perpendicular, turning the object.

15. In a curved path (non-circular), how do I find the total acceleration?

You usually have two parts: the Tangential Acceleration (which changes how fast you go) and the Radial/Centripetal Acceleration (which turns you). The total acceleration vector is the vector sum of these two components

Lesson 4: Projectile Motion: Why Horizontal & Vertical Motions are Independent

16. Why do we treat horizontal and vertical motion completely separately?

Because gravity only acts vertically. It pulls things down, not sideways. This means the horizontal acceleration is zero (constant speed), while the vertical acceleration is constant gravity (g). You can solve the x-problem and the y-problem as if they are two different worlds, linked only by the variable of time t.

17. At the very top of a projectile's path, is the velocity zero?

No! The vertical velocity is zero (it stopped going up and hasn't started falling yet), but the horizontal velocity is exactly the same as when it was launched. If the total velocity were zero, the object would drop straight down like a stone, not continue in an arc.

18. If I drop a bullet and shoot a bullet horizontally at the same time, which hits the ground first?

Assuming the ground is flat, they hit at the exact same time. The horizontal speed of the fired bullet does not cancel out gravity. Both bullets start with zero vertical velocity and fall the same height, so gravity pulls them to the ground in the same amount of time.

19. How do I derive the "Range Equation" R = (v² sin 2theta)/g and when am I allowed to use it?

This formula is derived by substituting the total flight time into the horizontal distance equation. Warning: You can only use this formula if the projectile lands at the exact same height it launched from (level ground). If you shoot off a cliff, this formula will fail.

20. Does the mass of the projectile affect how far or fast it travels?

In ideal physics problems (no air resistance), Mass cancels out of the equations entirely. A bowling ball and a ping pong ball launched at the same speed and angle will follow the exact same path.

Lesson 5: Uniform Circular Motion: Centripetal Acceleration & Period

21. Does "constant speed" in a circle mean zero acceleration?

No. Velocity is a vector (speed + direction). Even if speed is constant, your direction is constantly changing as you turn. Changing direction requires a force and an acceleration. This is Centripetal Acceleration.

22. Why is the formula for Centripetal Acceleration a = v²/r?

This comes from the geometry of the circle. As you move a tiny distance along the arc, your velocity vector changes direction by a tiny angle. If you use similar triangles to compare the position vectors and velocity vectors, the ratio yields v²/r.

23. Is there a force pushing me outwards (Centrifugal force)?

No, that is a "fictitious" force caused by your own inertia. Your body wants to keep moving in a straight line (Newton's 1st Law). The car turns inward, and your body feels like it's being thrown outward, but actually, the car door is pushing you inward to make you turn.

24. How do I calculate the Period (T) of an object in circular motion?

The Period is the time it takes to complete one full revolution. Since Distance = Speed * Time, for a circle, the distance is the circumference (2 * pi * r). So, 2 * pi * r = v * T. Rearranging this gives T = (2 * pi * r) / v.

25. If I swing a ball on a string and the string breaks, which way does the ball fly?

It flies off in a straight line tangent to the circle at the exact point where the string broke. It does not fly straight out away from the center, nor does it keep curving. It follows the velocity vector it had at that instant.