Position Needs a Coordinate Convention
A statement such as “the particle is 4 meters from the marker” gives a distance, but it does not say which side of the marker the particle is on. To describe a location unambiguously along a straight line, choose a reference point and a direction.
The reference point is the origin, assigned coordinate zero. The chosen direction is the positive direction. Locations in that direction receive positive coordinates; locations in the opposite direction receive negative coordinates. A coordinate also includes a unit, such as meters or kilometers.
These choices are part of the model. For example, on a straight road, one could choose east as positive or west as positive. Either choice can describe the same physical locations, but the coordinate signs will differ.
Position is not the same thing as “how far the object has traveled.” It identifies where the object is at a particular instant according to the selected coordinate system. A position of \(-4\) meters means the object is 4 meters from the origin on the negative side, not that it has traveled a total of 4 meters.
Position and Displacement Are Different
A single position describes one location. Displacement compares two locations: it is the signed change from an initial position to a final position. Both positions must be expressed using the same origin, positive direction, and units.
A positive displacement means the final coordinate is greater than the initial coordinate. A negative displacement means the final coordinate is less than the initial coordinate. These statements concern the change in position, not whether either position itself is positive or negative.
For example, an object can have a positive position and a negative displacement: it may begin at \(+12\) meters and end at \(+4\) meters. It remains on the positive side of the origin, but its coordinate has decreased. Conversely, an object can begin at a negative position and have positive displacement by ending at a greater coordinate.
Why Displacement Is Final Position Minus Initial Position
On a number line, coordinates record signed locations from one fixed origin. The signed change needed to go from coordinate \(x_i\) to coordinate \(x_f\) is the amount that, when added to \(x_i\), gives \(x_f\). Calling that change \(\Delta x\), we have \(x_i+\Delta x=x_f\). Subtracting \(x_i\) from both sides gives the displacement formula.
This reasoning also explains why a negative initial position must be subtracted with its sign intact. If \(x_i=-3\), then moving from \(x_i\) to \(x_f=5\) requires the change \(5-(-3)\), not \(5-3\). The displacement carries the object from its initial coordinate to its final one.
Worked Examples
Worked Example: Moving from South of a Marker to North
A straight path runs north–south. Let a marker be the origin, and choose north as the positive direction. A person is initially \(3\) kilometers south of the marker and later is \(5\) kilometers north of it. Find both positions and the displacement.
Step 1: Assign signed positions. South is opposite the positive direction, so the initial position is \(x_i=-3\text{ km}\). North is the positive direction, so the final position is \(x_f=+5\text{ km}\).
Step 2: Subtract initial position from final position.
The displacement is \(+8\) kilometers, meaning a net change of 8 kilometers north under this convention. Check by adding the displacement to the initial position:
The check confirms the result.
Worked Example: Positive Positions but Negative Displacement
A cart has initial position \(+12\) meters and final position \(+4\) meters on a track whose positive direction points to the right. Find its displacement and interpret the signs.
Step 1: Identify the two coordinates. The cart starts at \(x_i=+12\text{ m}\) and ends at \(x_f=+4\text{ m}\). Both positions are positive, so both locations are to the right of the origin.
Step 2: Calculate the displacement.
The displacement is \(-8\) meters. Its negative sign means the final coordinate is 8 meters less than the initial coordinate, a net change opposite the chosen positive direction. The negative displacement does not mean that either position is negative: both positions are positive.
Check:
The initial position plus the displacement equals the final position, as required.
Worked Example: A Negative Position and a Positive Displacement
A point on a number line begins at \(-6\) centimeters and ends at \(-2\) centimeters. The positive direction is to the right. Find its displacement and explain why it is positive even though both positions are negative.
Step 1: Record the signed coordinates. The initial position is \(x_i=-6\text{ cm}\), and the final position is \(x_f=-2\text{ cm}\). Both points lie to the left of the origin.
Step 2: Calculate final minus initial.
The displacement is \(+4\) centimeters because the final coordinate is greater than the initial coordinate. The point ends closer to zero, but the key sign test is the comparison \(-2>-6\). A positive displacement does not require a positive final position.
Check:
The result is consistent with both given positions.
Coordinates Depend on the Chosen Origin and Direction
The same physical location can have different coordinates under different conventions. Moving the origin changes the numerical positions. Reversing the positive direction changes their signs relative to the new direction. Therefore, a coordinate is meaningful only when its origin, direction, and unit are understood.
Suppose an old coordinate is \(x\), and a new origin is placed at old coordinate \(a\). If the positive direction is unchanged, the new coordinate is \(x'=x-a\): subtracting \(a\) makes the new origin have coordinate zero. If the positive direction is reversed as well, the new coordinate is \(x'=-(x-a)\). More generally, write \(s=+1\) for the same orientation and \(s=-1\) for the reversed orientation. Then \(x'=s(x-a)\).
This change also shows how displacement responds to a new convention. Apply the coordinate rule to the final and initial positions and subtract:
When only the origin changes, \(s=+1\), so the displacement stays the same. When the orientation is reversed, \(s=-1\), so the displacement changes sign. This is why a displacement’s sign must always be interpreted using the chosen positive direction.
Worked Example: Reversing the Positive Direction
With east chosen as positive, a point moves from \(2\) kilometers east of an origin to \(7\) kilometers east of it. Find the displacement using east-positive coordinates, then express the same two locations relative to a new origin \(1\) kilometer east of the old one, with west chosen as positive.
Step 1: Calculate displacement in the original coordinates. The initial and final positions are \(x_i=+2\text{ km}\) and \(x_f=+7\text{ km}\).
Step 2: Find the new positions. The new origin is at \(a=+1\text{ km}\) in the old coordinates. Since the new positive direction is west, \(s=-1\), so \(x'=-(x-a)\). For the initial location,
For the final location,
Step 3: Calculate displacement in the new coordinates.
The displacement is now \(-5\) kilometers because west is positive and the net change is eastward. Reversing the orientation reverses the sign. Check the new-coordinate calculation:
The arithmetic is consistent, and the physical locations have not changed.
Common Sign Mistake
A frequent error is to treat a negative initial coordinate as if it were a positive distance when subtracting. For an object that starts at \(-4\) meters and ends at \(+3\) meters, the incorrect calculation might be
This reverses the roles of the initial and final positions and does not satisfy the defining check. The correct calculation is final position minus initial position:
Check the corrected result:
The positive sign indicates a net change in the chosen positive direction. It does not mean that the starting position was positive; the starting position was \(-4\) meters.
Keep the Three Ideas Separate
When describing a one-dimensional location or change, identify which quantity is being discussed before interpreting its sign:
- Position answers “where is the object relative to the origin?”
- Displacement answers “what is the signed change from the initial position to the final position?”
- Positive direction specifies how the signs of both positions and displacements are interpreted.
A reliable procedure is to state the origin and positive direction, assign signed coordinates with units, calculate displacement as final minus initial, and check that the initial position plus the displacement equals the final position. This prevents the sign of a location from being confused with the sign of a change.
Check Your Understanding
Use a clearly stated origin and positive direction when needed. For each displacement calculation, check whether adding the displacement to the initial position gives the final position.
- A person is \(6\) meters west of a marker and later \(2\) meters east of it. If east is positive, what are the two positions and the displacement?
- An object begins at \(+9\) centimeters and ends at \(+1\) centimeter. Can its displacement be negative even though both positions are positive? Calculate it.
- An object moves from \(-8\) meters to \(-3\) meters. What is its displacement, and why can that displacement be positive while the final position remains negative?
- In the formula \(\Delta x=x_f-x_i\), what mistake occurs if \(x_i\) is negative and its negative sign is dropped before subtraction?
- If the origin is shifted but the positive direction stays the same, does the displacement change? Explain using the coordinate transformation.
