Terminology
Position is a location at a specific time. Position is usually represented as x
Distance is a scalar quantity that refers to how much an object travelled over a specific time.
Displacement is the overall change in position of an object over a specific time, including direction. Displacement is calculated by xfinal - xinitial and represented by the symbol delta x.
Distance is a scalar quantity that refers to how much an object travelled over a specific time.
Displacement is the overall change in position of an object over a specific time, including direction. Displacement is calculated by xfinal - xinitial and represented by the symbol delta x.
Interpreting Position Time graphs
Consider a car moving constantly to the right (positive direction +) with a constant velocity of 5m/s
(figure 1)
The position-time graph of this car would be:
A motion described with a constant, positive velocity results in a line with positive, constant slope on the position-time graph.
The y-intercept of a position-time graph is the starting position. The slope of a position-time graph is the velocity of an object
If the slope is positive, the velocity is positive. If the slope is negative, the velocity is negative. If the slope is shallow, then the velocity is low. If the slope is steep, the velocity is high.
The speed of an object is the magnitude of its velocity. Thus, speed in a position-time graph is the steepness, or absolute value of, the slope.
Position-time graph of an object under uniform acceleration
In another example, a car is accelerating constantly in the positive direction. The velocity of the car is changing and the car is moving faster and faster.
(figure 2)
The position-time graph of this motion would be like the below, with slope increasing over time.
The object is moving to the positive direction with a positive acceleration. The signs are the same, suggesting that it is speeding up. If the sign of the velocity is different than the sign of acceleration, then the object is slowing down. The graph below shows an object moving in the positive direction with a negative acceleration, which is slowing down.
This applies to objects moving in a negative direction as well:
Determining acceleration and starting velocity from a Position Time Graph
There are two ways to find acceleration of an object from a position time graph, One way is to use a graphic calculator and make a line of best fit of a quadratic model to the data given. The line would be in the form of ax^2+bx+c, where a is the acceleration. Another way to find acceleration is to calculate the velocity over each time interval and calculate the change in velocity. If the object is under uniformly accelerated motion, the change in velocity (acceleration) should be the same in each time interval. The starting velocity of an object is the slope at time zero in a position time graph. This can be best calculated by dividing the change in position (displacement) with a small time interval near zero such as (0,0.1).
Interpreting Velocity-Time Graphs
Velocity-time graphs plots the velocity of an object over time. In the first example of the previous section, the car is moving with a positive, constant velocity, thus the velocity time graph is a horizontal straight line with a slope of zero above the time axis:
However, if the slope of the line in a velocity time graph is not zero, then the object is accelerating. The velocity time graph of the car moving with constant, positive acceleration in the second example of the previous section would be a linear line with a positive slope.
The y-intercept of a velocity-time graph is the initial velocity. The slope of a velocity-time graph is the acceleration of an object
We can make a similar analysis of speed in the previous section with velocity time graphs. In a velocity-time graph, if the slope of the line is positive and above the time axis, the object is speeding up. If the slope is positive but below the time axis, the object is slowing down. If the slope is negative and below the time axis, the object is speeding up. If the slope is negative and above the time axis, the object is slowing down. In addition, whenever the line in a velocity-time graph crosses, the time axis, the object changes direction.
Determining Position, Distance and Displacement from a Velocity Time Graph
Displacement of an object is the net area under the velocity-time graph. In the example below, it is the blue area. Distance is the total area under the curve. In the example below, it is also the blue area.
However, in this example where the line crosses the time axis, the displacement is the blue area minus the red area (the net area). On the other hand, the distance in this example is the blue area plus the red area (total area).
You can find the position of an object at specific times with the velocity-time graph if give the initial position of the object. The position of the object at time t would be the initial position plus the area under the line of a velocity-time graph between zero and t.
Connecting position-time graphs, velocity-time graphs and strobe diagrams
Velocity-time graph is the derivative of the position time graph. If the position-time graph is quadratic, the velocity time graph is linear. If the position-time graph is linear, the velocity time graph is flat. In strobe diagrams like figure 1 and figure 2, if the object moves the same distance every second, it has a constant velocity. If the object moves at increasing increments every second, it has a positive acceleration.
Solving Problems
There are five variables in kinematics problems:
1. xinitial (initial position)
2. xfinal (final position)
3. vinitial (initial velocity)
4. vfinal (final velocity)
5. a (acceleration)
Using the 4 kinematic equations, if you know 3 of the variables, you can find the other two.
1. xinitial (initial position)
2. xfinal (final position)
3. vinitial (initial velocity)
4. vfinal (final velocity)
5. a (acceleration)
Using the 4 kinematic equations, if you know 3 of the variables, you can find the other two.
Projectile Motion
A projectile is an object moving in the air and the only force acting upon it is gravity.
- Horizontal motion determines the distance object travels
- Vertical motion determines the time object stays in the air
The time that object stay in the air is same for vertical and horizontal motion
- Horizontal motion determines the distance object travels
- Vertical motion determines the time object stays in the air
The time that object stay in the air is same for vertical and horizontal motion
A, B, and C are trajectories of three projectiles that start at the same position. The time that A, B, and C are in the air is the same since they reach the same highest point and their vertical motion is the same. However, the starting velocity of C is greater than B which is greater than A. Since the horizontal motion determines the distance an object travels, A travels the shortest distance in the same amount of time as B and C, thus having the lowest initial velocity.
A, B, and C are trajectories of three projectiles that start at the same position. The initial velocities of A, B, and C are the same since they have the same horizontal motion. However, the time that they stay in the air is different since they reach different heights and their vertical motion is different, with C reaching the highest point and staying in the air the longest.