2.1 Displacement - College Physics | OpenStax

Position

To describe the motion of an object, its position—where it is at any particular time—must be specified relative to a convenient reference frame. Earth is frequently used as a reference frame, such as describing a rocket launch relative to Earth or a professor's position relative to a whiteboard. Reference frames can also be in motion relative to Earth, such as using an airplane as a reference frame to describe a passenger's position. Objects such as cyclists in Vietnam can also be described relative to reference structures like buildings and a canal.

Displacement

If an object moves relative to a reference frame, its position changes. This change in position is defined as displacement:

Δx=xf−x0\Delta x = x_f - x_0

In this equation, Δx\Delta x represents displacement, xfx_f represents final position, and x0x_0 represents initial position. The upper case Greek letter Δ\Delta (delta) means "change in" whatever quantity follows it. Displacement is always calculated by subtracting the initial position x0x_0 from the final position xfx_f.

The SI unit for displacement is the meter (m\text{m}), although other units such as kilometers (km\text{km}), miles (mi\text{mi}), or feet (ft\text{ft}) are sometimes used. When non-SI units are used, conversion to meters may be necessary for calculations.

Displacement has both direction and magnitude. In one-dimensional motion, direction is indicated using a positive (++) or negative (−--) sign relative to a chosen coordinate system:

  • Professor Example: Relative to Earth, a professor has an initial position x0=1.5 mx_0 = 1.5\,\text{m} and a final position xf=3.5 mx_f = 3.5\,\text{m}. Her displacement is:

Δx=xf−x0=3.5 m−1.5 m=+2.0 m\Delta x = x_f - x_0 = 3.5\,\text{m} - 1.5\,\text{m} = +2.0\,\text{m}

The positive value indicates motion to the right.

  • Airplane Passenger Example: Relative to an airplane, a passenger moves toward the rear from an initial position x0=6.0 mx_0 = 6.0\,\text{m} to a final position xf=2.0 mx_f = 2.0\,\text{m}. His displacement is:

Δx=xf−x0=2.0 m−6.0 m=−4.0 m\Delta x = x_f - x_0 = 2.0\,\text{m} - 6.0\,\text{m} = -4.0\,\text{m}

The negative value indicates motion toward the rear of the airplane (the negative direction in the coordinate system). The displacement arrow for the passenger is twice as long as that for the professor because the passenger moved twice as far.

Distance and Distance Traveled

While displacement includes direction, distance does not:

  • Distance: The magnitude or size of displacement between two positions. It has no direction and no sign.

  • Distance Traveled: The total length of the path traveled between two positions.

Distance traveled can be greater than the magnitude of displacement. For example, a professor pacing back and forth during a lecture might travel a distance of 150 m150\,\text{m}, while her overall displacement is +2.0 m+2.0\,\text{m} and the magnitude of her displacement is 2.0 m2.0\,\text{m}.

Displacement depends only on the start and end positions and is independent of the path taken. Distance traveled depends on the total path taken between those points. Kinematics deals almost exclusively with displacement and magnitude of displacement.

Example Problem: Cyclist Motion

A cyclist rides 3 km3\,\text{km} west and then turns around to ride 2 km2\,\text{km} east. Taking east as positive and west as negative:

  • (a) Displacement:

Δx=xf−x0=−1 km\Delta x = x_f - x_0 = -1\,\text{km}

The displacement is negative because net motion is toward the west.

  • (b) Distance Traveled:

Distance=3 km+2 km=5 km\text{Distance} = 3\,\text{km} + 2\,\text{km} = 5\,\text{km}

  • (c) Magnitude of Displacement:

Magnitude=1 km\text{Magnitude} = 1\,\text{km}