For non-physicists, maps and speedometers come in handy when assessing a change in position or a change in speed of an object. But when you’re a physicist, graphs – graphs of motion in particular – are super important for determining the position or the rate of change of speed of an object. As you’ll see below, graphs of motion help us, as physics students, better understand the movement of a body over a certain period.
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Jetzt kostenlos anmeldenFor non-physicists, maps and speedometers come in handy when assessing a change in position or a change in speed of an object. But when you’re a physicist, graphs – graphs of motion in particular – are super important for determining the position or the rate of change of speed of an object. As you’ll see below, graphs of motion help us, as physics students, better understand the movement of a body over a certain period.
There are three main types of graphs used to define the motion of an object in a straight line: displacement-time graphs, velocity-time graphs, and acceleration-time graphs.
Figure 1 illustrates a displacement-time graph of an object moving at a constant velocity. For the displacement-time graph, displacement (denoted by d) is on the y-axis, and time (denoted by t) is on the x-axis.
From such a graph, we can obtain
To calculate the slope p of the above graph, we use the following equation:
\[p = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta d}{\Delta t} m/s\]
The rate of change of displacement is velocity, so the slope of the displacement-time graph is the velocity.
Have a look at the velocity-time graph below:
For the velocity-time graph, velocity (v) is on the y-axis, and time (t) is on the x-axis. From this graph, we can find
To calculate the slope p of the above graph, we use the following equation:
\[p = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta v}{\Delta t} m/s^2\]
The rate of change of velocity is acceleration, so the slope of the velocity-time graph is the acceleration.
Furthermore, the area under the velocity-time graph gives the distance covered by the object, which is the displacement.
For an acceleration-time graph, acceleration (a) is given on the y-axis and the time (t) on the x-axis. The acceleration-time graph gives us the acceleration at any given time. Also, the area under the acceleration-time curve represents a change in velocity.
Below we explore how to draw graphs of motion for different scenarios.
For an object at rest, there will be no change in displacement, which will result in no change in velocity, and because there is no change in the velocity, the change in acceleration will be zero as well.
An object at rest will not move. Hence, the displacement will not change over the interval of time which is depicted by a flat line parallel to the time axis.
The velocity will be zero because the object’s displacement doesn’t change. Hence, the graph for an object without its velocity changing over time can be shown with a straight line on the time axis.
The acceleration will be zero because the object’s velocity is not changing and the acceleration time graph be a flat line starting from the origin.
When an object moves at a constant velocity:
The slope of the below graph is positive, indicating the movement is in the positive direction (away from the origin). If this curve had been the same but with a negative gradient (towards the origin), it would have depicted displacement in the opposite direction. Also, displacement is uniformly increasing because the velocity is constant.
Question: Which direction should be considered positive or negative?
Answer: The sign is arbitrary. You can take any direction as either positive or negative.
As the slope of the displacement-time graph for a body moving with a constant velocity is positive in figure 7, the velocity is a constant straight line in the positive direction.
As there is no rate of change of velocity (constant velocity), the acceleration will be zero as well because for acceleration or deceleration to occur, there needs to be a change in velocity as well.
When an object moves with constant acceleration:
Below are the two graphs for displacement vs time. Figure 10 is for constant acceleration and figure 11 is for constant deceleration.
If you take the tangents at various points on both of the above curves, you will see that the slope of the displacement-time graph in figure 10 is becoming steeper and steeper. This is an indication that the velocity is increasing. In figure 11, the gradient is gradually decreasing, which is an indication that the velocity is decreasing.
The velocity-time graph for a constant acceleration will be a uniformly increasing line as shown by the figure below.
As the acceleration is not changing over time and is constant, the acceleration-time graph can be represented by a straight line.
When an object moves with a constant deceleration:
Because the body is decelerating, the curve is approaching a constant (unchanged) value.
The velocity-time graph for a constant deceleration will be a uniform line constantly decreasing from some value.
The constant line with a negative acceleration shows that an object is decelerating with a constant value.
For this scenario, an object, let’s say a ball, is thrown upwards in such a way that it lands in the thrower’s hand after some time. Air resistance is negligible, and the only forces acting on the ball come from the thrower (to throw the ball upwards) and the gravitational pull on the ball until it lands in the thrower’s hand. The upwards direction is considered as positive.
The displacement-time graph for an object thrown straight up and then landing in the thrower's hand is shown below.
Once the ball is thrown in the air, the ball’s displacement increases because we have taken the upwards direction as positive. As it reaches the top, the gradient of the displacement-time graph will be zero for a brief moment, indicating that the ball is changing its direction and will move downwards from here on. Therefore, the graph will move downwards until the ball reaches its original position.
But why is the graph a curve and not a straight line? The acceleration due to gravity is constant, with a value of 9.81m/s2. So, from the moment the ball is thrown until it is caught, the deceleration due to gravity and the acceleration due to free-fall will be constant and different from zero.
The velocity-time graph for an object thrown straight up and then landing in the thrower's hand is shown below.
The ball is thrown upwards with some initial velocity u. As the ball reaches the top, its velocity decreases uniformly until it reaches zero, where the ball is at rest for a brief moment. Afterwards, the ball moves downwards with a uniformly increasing velocity.
As the distance travelled will be the same upwards and downwards because of negligible air resistance, the initial velocity will be equal to the final velocity -u. So, the area of both regions A and B will be the same in this case.
Why is the slope of the graph negative and not positive after u reaches zero? As the upwards direction is taken as positive, once the direction of the ball changes at the top, the motion will be downwards in the negative direction with a constant acceleration of free fall.
The acceleration-time graph for an object thrown straight up and then landing in the thrower's hand is shown below.
The acceleration is a constant -9.81m/s2 throughout the displacement as the velocity-time graph is uniformly decreasing.
After the ball is tossed in the air, the gravitational force works in the direction opposite to the upwards motion. Since the upwards motion is taken to be positive, the gravitational force will be negative. Once the ball reaches its peak, the ball changes direction. Hence, the gravitational force will continue to be negative.
There are three main types of graphs concerning linear motion: the displacement-time graph, the velocity-time graph, and the acceleration-time graph
You can interpet graphs of motion as follows: The gradient of the graph and area under the graph give information about the displacement, velocity, and acceleration depending on what type of linear motion graph we are considering. For example, the slope of a displacement-time graph gives the average velocity.
The graphs of motion are the displacement-time graph, the velocity-time graph, and the acceleration-time graph.
Graphs of motion are used to display linear motion. The graphs of motion are the displacement-time graph, the velocity-time graph, and the acceleration-time graph.
You can work out the velocity of motion graphs as follows:
On a distance-time graph, what does a steeper line represent?
The movement is faster.
What does a straight, horizontal line on a velocity-time graph represent?
The speed of the moving object is constant.
Suppose you have a curve on a speed-time graph, and the curve is going upwards. What can you deduce from this?
The moving body is accelerating.
On a distance-time graph, what can you deduce if there is a horizontal line?
The body is stationary.
Distance moved in a specified direction is referred to as _____?
Displacement.
What does a steeper line on a speed-time graph represent?
The acceleration will be faster.
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