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. Show
Types of motion graphsThere 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. Displacement versus time graphFigure 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:
The rate of change of displacement is velocity, so the slope of the displacement-time graph is the velocity. Velocity versus time graphHave 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: 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. Acceleration versus time graphFor 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.
Analyzing graphs of motion without numbersBelow we explore how to draw graphs of motion for different scenarios. Graphs of Motion: An object at restFor 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. Displacement-time graph for an object at restAn 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.
Velocity-time graph for an object at restThe 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.
Acceleration-time graph for an object at restThe 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:
Displacement-time graph for an object moving at a constant velocityThe 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. Velocity-time graph for an object moving at a constant velocityAs 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.
Acceleration-time graph for an object moving at a constant velocityAs 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.
Graphs of Motion: Objects moving with a constant acceleration (distance is increasing over time)When an object moves with constant acceleration:
Displacement-time graph for an object moving with constant accelerationBelow 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.
Velocity-time graph for an object moving with constant accelerationThe 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.
Graphs of Motion: An object moving at a constant deceleration (distance is increasing over time)When an object moves with a constant deceleration:
Displacement-time graph for an object moving with constant decelerationBecause the body is decelerating, the curve is approaching a constant (unchanged) value.
Velocity-time graph for an object moving with constant decelerationThe velocity-time graph for a constant deceleration will be a uniform line constantly decreasing from some value.
Acceleration-time graph for an object moving with constant decelerationThe constant line with a negative acceleration shows that an object is decelerating with a constant value.
Graphs of Motion: Throwing an object straight up with the object returning to the throwerFor 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. Displacement-time graph for an object thrown straight upThe 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. Acceleration-time graph for an object thrown straight upThe 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. Graphs of Motion - Key takeaways
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