3.5 Position-Time Graphs

Definition

A position-time graph shows how the position of an object changes with time.

The horizontal axis represents time, and the vertical axis represents position relative to a chosen origin. Each point on the graph gives the position of the object at a particular instant.

Figure 3.5.1 – A position-time graph showing the position of an object at different times.

For example, the point (3 s, 6 m) means that the object is at position x = 6 m at t = 3 s.

Reading a Position-Time Graph

A position-time graph describes both where an object is and how its position changes with time.

Consider the graph in Figure 3.5.1.

At t = 0 s, the object is at x = 0 m.

At t = 2 s, the object is at x = 4 m.

At t = 4 s, the object is at x = 8 m. The displacement between any two times can be found from the corresponding positions:

Δx=xf−xi\Delta x = x_f – x_i

For example, between t = 1 s and t = 4 s,

Δx=8m−2m=+6m\Delta x = 8\,\mathrm{m} – 2\,\mathrm{m} = +6\,\mathrm{m}

The positive displacement indicates that the object’s position has changed in the positive x-direction.

Slope of a Position-Time Graph

The slope of a position-time graph represents velocity.

For two points on the graph,

slope=change in positionchange in time\text{slope}=\frac{\text{change in position}}{\text{change in time}}

Therefore,

v→avg=Δx→Δt\vec{v}_{\mathrm{avg}}=\frac{\Delta \vec{x}}{\Delta t}

where

v→avg=average velocity\vec{v}_{\mathrm{avg}}=\text{average velocity}

Δx→=displacement\Delta \vec{x}=\text{displacement}

Δt=time interval\Delta t=\text{time interval}

For the straight-line graph in Figure 3.5.1,

v = (8 m − 0 m) / (4 s − 0 s)

v = +2 m/s

The slope is constant, so the velocity is constant.

A steeper slope represents a velocity with a greater magnitude.

Positive, Negative, and Zero Slope

The sign of the slope indicates the direction of motion.

Figure 3.5.2 – Positive, zero, and negative slopes on position-time graphs.

A positive slope means that position increases with time. The velocity is positive.

A negative slope means that position decreases with time. The velocity is negative.

A zero slope means that position does not change with time. The velocity is zero, so the object is at rest.

The sign of the position itself does not determine the direction of motion. The direction is determined by how the position changes with time.

Changing Velocity

If the slope of a position-time graph changes, the velocity is changing.

A straight line has a constant slope and therefore represents constant velocity.

A curved line has a changing slope and therefore represents changing velocity.

Figure 3.5.3 – A curved position-time graph has a changing slope, indicating changing velocity.

In Figure 3.5.3, the slope becomes progressively steeper. The velocity is therefore positive and increasing in magnitude.

Average and Instantaneous Velocity from a Position-Time Graph

For motion with changing velocity, the slope between two points gives the average velocity over that time interval.

Figure 3.5.4 – The slope of the secant line between two points gives the average velocity over that time interval.

The average velocity between points A and B is

vavg=ΔxΔtv_{\mathrm{avg}}=\frac{\Delta x}{\Delta t}

The instantaneous velocity at a particular time is given by the slope of the tangent to the position-time graph at that point.

Figure 3.5.5 – The slope of the tangent at a point gives the instantaneous velocity at that instant.

A steeper tangent corresponds to a greater magnitude of instantaneous velocity.

Example: Interpreting a Position-Time Graph

Figure 3.5.6 – Position-time graph for an object during three different intervals of motion.

From A to B,

v = (8 m − 0 m) / (4 s − 0 s) = +2 m/s

The object moves in the positive x-direction with constant velocity.

From B to C,

v = (8 m − 8 m) / (7 s − 4 s) = 0 m/s

The position remains constant, so the object is at rest.

From C to D,

v = (0 m − 8 m) / (11 s − 7 s) = −2 m/s

The object moves in the negative x-direction with constant velocity.

The graph therefore shows that the object moves away from the origin, remains at rest for a period of time, and then returns to the origin.

Key Points

  • A position-time graph shows how an object’s position changes with time.
  • Time is plotted on the horizontal axis and position on the vertical axis.
  • Each point on the graph represents the object’s position at a particular instant.
  • The slope of a position-time graph represents velocity.
  • A positive slope represents positive velocity.
  • A negative slope represents negative velocity.
  • A horizontal line represents zero velocity and an object at rest.
  • A straight line represents constant velocity.
  • A curved line represents changing velocity.
  • The slope between two points gives average velocity.
  • The slope of a tangent gives instantaneous velocity.
  • The sign of position does not determine the direction of motion.

← 3.4 Acceleration 

3.6 Velocity-Time Graphs →