Graphical Analysis (slope)
Lab: Velocity and Acceleration
Students will have collected data relating to their cart’s motion as shown below:
1 m / 2 m / 3 m / 4 m / 5 mAvg. Time / 1.1 s / 2.5 s / 4.3 s / 6.7 s / 10 s
Velocity (speed) / 0.9 m/s / 0.8 m/s / 0.7 m/s / 0.6 m/s / 0.5 m/s
Acceleration / -0.07 m/s2 / -0.05 m/s2 / -0.04 m/s2 / -0.03 m/s2
Average time: to calculate the average time, add all the measured times and divide by the number of measurements. Ex.:
Time to 1 meter1.2 s
1.0 s
1.1 s
3.3 s / Total
1.1 s / Divided by 3
Velocity (speed in a given direction): to calculate the velocity we use the formula, where x refers to position and t refers to time. The starting point of the cart’s measurement was at zero meters (bottom of the ramp). This is the initial position or xi. The cart moved 1 m (this is the final position or xf) for a length of time of 1.1 s. Therefore the above equation becomes:
The same calculation can be done between the initial position of zero meters and the final position of 2 meters; producing a velocity of 0.75 m/s.
Acceleration: to calculate acceleration we must examine the data and realize that we can only calculate the acceleration between 1 and 2 meters, between 2 and 3 meters, and so on…The formula for acceleration is. [In order to use this formula we are assuming that time measurements were done simultaneously. In other words, we assume that the cart was allowed to travel 5 m while all the time measurements were made as the cart reached each set position. (Having made several time measurements for each length the car moved, we can safely say that the time measurements collected reflect the continuous motion the car made)]. With this in mind we can substitute the numbers on the formula and find that the acceleration between 1 m and 2 m is:
Notice the acceleration is negative. This is correct and coincides with the decrease in speed. The decrease in speed is related to the friction that the wheels have against the floor and with the axles of the cart. (This example can be used when discussing forces and simple machines).
Graphing: In order to begin the graph several things must be analyzed: the purpose of the graph, the dependent and independent variables, the frame of reference and scale, and the units used to make the measurements.
- Dependent and independent variables:
Students need to be able to recognize which measurements (variables) were manipulated and which measurements depended on said manipulations. In this experiment, the length that the cart traveled was manipulated and selected prior to recording the time. Therefore, the time measured depended on the set positions of the cart. This translates to position being the independent variable, and the dependent variable being time.How do we identify the position of variables on a graph? As a rule of thumb always remember this basic equation y = f(x); which describes that y is a function of x; or more applicable to our question, y depends on x. The variable y is the Y-Axis, and the variable x is the X-Axis.
- Purpose of the graph:
The purpose of the graph is to create a visual representation of the motion of the cart and to examine how the motion of the cart is dependent on its change in position and the time required to make such change. Because of the types of analyses required to make on graphs describing rates of change, time should always be drawn on the X-Axis, with the understanding that the information graphed will not change. (see graphs)
- Frame of reference:
In this activity the frame of reference was chosen with the bottom of the ramp to be the starting position and the point where time was zero. All measurements were made from the bottom of the ramp. The graph will then have the origin at zero seconds and zero meters. The scales will depend on the highest number for each measurement. Since the highest time measurement was 10 seconds, the scale must reach at least to 10 seconds. The highest distance measured was 5 meters.
- Units used to measure:
The units used to measure time were seconds (with tenthsof second as the smallest measurement), and the units of position (distance) were meters. This means that the time scale must start at zero seconds and have an accuracy of a tenth of a second, while the distance scale must have an accuracy of a meter.
Graph 1: Time vs. Position (time is dependent of position)
Graph 2: Position vs. Time (Position is dependent of time)
Looking at both graphs we see that the position of the car is increasing as time increases, but the change in position per given time is decreasing (it takes more time to travel each meter). The second graph is easier to analyze because by definition speed (velocity) is distance (displacement) over time and to find the slope of a line we must divide the change in Y-Axis over the change in X-Axis.
The equation of a line is:
where y refers to the Y-Axis and x to the X-Axis, and m is the slope
Rearranging the equation we find that slope is:
In the second graph the slope is given by this equation:
In this graph the numbers represent the slopes used to calculate the speeds between each meter. As can be seen the slopes are decreasing, hence the decrease in speed.
Extension:
Additional work can be done by graphing the calculated velocities vs. time. Unfortunately, the changes in velocities are not exact and the slopes do not give a constant acceleration, which is expected. To find an approximate acceleration a slope of the whole curve is done (red line below) which provides a slope of - 0.04 m/s2, a very close approximation to the uniform acceleration of the cart.