KAHIBARO
Table of Contents
Reading Position-Time Graphs
A position-time graph shows how an object's position changes as time passes. On this graph, time is placed on the horizontal axis, and position is placed on the vertical axis. Each point on the graph tells you where the object is at a particular instant.
If the motion is along a straight line, the position may be positive, negative, or zero, depending on the chosen origin. A position-time graph does not directly show force or acceleration. It shows position, and from its shape we can infer other features of motion.
What the Graph Represents
Suppose an object moves along the $x$ axis. Then a position-time graph plots $x$ versus $t$. If at time $t = 0$ the object is at $x = 2 \, \text{m}$, then the graph begins at the point $(0, 2)$.
For motion with constant acceleration, the position changes according to
$$
x(t) = x_0 + v_0 t + \frac{1}{2}at^2
$$
This means the graph is generally curved, not straight, because of the $t^2$ term.
For constant acceleration, the position-time relation is
$$
x(t) = x_0 + v_0 t + \frac{1}{2}at^2
$$
A position-time graph for this motion is usually a parabola.
Slope and Its Meaning
The slope of a position-time graph tells us the velocity. A steep positive slope means the object is moving in the positive direction بسرعة كبيرة, that is, with large positive velocity. A negative slope means the object is moving in the negative direction.
At one point on a curved graph, the slope of the tangent line gives the instantaneous velocity. When the motion has constant acceleration, this slope changes steadily with time.
If the slope becomes larger and larger in the positive direction, the object is speeding up in the positive direction. If the slope becomes less positive and eventually negative, the object may slow down, stop, and reverse direction.
The slope of a position-time graph is velocity.
Average velocity between two times:
$$
v_{\text{avg}} = \frac{\Delta x}{\Delta t}
$$
Instantaneous velocity is the slope of the tangent line to the graph at one point.
Shape of the Graph for Constant Acceleration
Because the equation contains $t^2$, the graph bends. The direction of the bending depends on the sign of the acceleration.
If $a > 0$, the graph is concave upward. The slope increases with time.
If $a < 0$, the graph is concave downward. The slope decreases with time.
This curvature is the visual sign of acceleration on a position-time graph. A straight line means constant velocity, while a curved line means velocity is changing.
Common Cases
The table below summarizes the main shapes for constant acceleration.
| Acceleration | Initial velocity | Shape of position-time graph | What happens to slope |
|---|---|---|---|
| $a = 0$ | any constant value | straight line | remains constant |
| $a > 0$ | $v_0 > 0$ | curves upward | increases positively |
| $a > 0$ | $v_0 < 0$ | curves upward | may start negative, become zero, then positive |
| $a < 0$ | $v_0 > 0$ | curves downward | may start positive, become zero, then negative |
| $a < 0$ | $v_0 < 0$ | curves downward | becomes more negative |
Turning Points
A turning point on a position-time graph is a point where the object changes direction. At that instant, the slope of the graph is zero, so the velocity is zero.
For constant acceleration, this happens when
$$
v = v_0 + at = 0
$$
Solving for the time gives
$$
t = -\frac{v_0}{a}
$$
At this time, the graph has a highest point or a lowest point, depending on the sign of the acceleration.
A turning point occurs when velocity is zero.
For constant acceleration:
$$
v_0 + at = 0
\quad \Rightarrow \quad
t = -\frac{v_0}{a}
$$
On the position-time graph, this is where the tangent is horizontal.
Example of Upward Curving Motion
Consider
$$
x(t) = 1 + 2t + t^2
$$
Here, $x_0 = 1$, $v_0 = 2$, and $a = 2 \, \text{m/s}^2$ because $\frac{1}{2}a = 1$.
The graph starts at $x = 1$ when $t = 0$. Its slope is positive at first and becomes more positive as time increases. So the object moves in the positive direction and speeds up.
Example of a Direction Change
Consider
$$
x(t) = 5 + 4t - t^2
$$
Then $\frac{1}{2}a = -1$, so $a = -2 \, \text{m/s}^2$. The slope starts positive, but because acceleration is negative, the slope decreases with time. Eventually the slope becomes zero, and later becomes negative. This means the object first moves in the positive direction, stops for an instant, and then moves back.
Visual Interpretation
A position-time graph can answer questions such as where the object started, whether it moves forward or backward, whether it stops and turns around, and whether its motion is speeding up or slowing down.
It is important to remember that a high point on the graph does not mean high speed. It means large position. Speed is related to the steepness of the graph, not the height.
Do not confuse position with velocity.
On a position-time graph:
The vertical value gives position.
The slope gives velocity.
The curvature shows changing velocity, which means acceleration.
Sketch of Typical Position-Time Curves
The blue curve bends upward, showing positive acceleration. The red curve bends downward, showing negative acceleration. The marked point on the red curve is a turning point, where the slope is zero.
Comparing Straight and Curved Graphs
A straight position-time graph means the object has constant velocity. A curved one means the velocity changes. In this chapter, because the motion has constant acceleration, the curve is smooth and parabolic.
This makes position-time graphs especially useful for recognizing whether acceleration is present and whether the object changes direction during its motion.
Final Idea
For motion with constant acceleration, the position-time graph gives a picture of how position evolves in time. Its starting point shows the initial position, its slope shows velocity, and its curvature reflects acceleration. By learning to read the shape of the graph, you can understand the motion without even looking at the full equation.