A student uses a sonic motion detector to determine the position of a car at various times. The student takes five positions at five different times. The student's data is represented below:
Time t [s]
Position x [m]
0.5
39.9
1.4
75.4
2.9
139.3
3.7
186.5
4.1
193.2
(a) Plot the data so that the points will yield a line whose slope is the car’s velocity. Be sure to label axes and indicate scale.
(b) Determine the velocity of the car in [m/s] using least-squares regression. Round to three decimal places
To plot the data and determine the velocity of the car using least-squares regression, you can follow these steps:
(a) Plotting the Data:
1. Create a graph with time (t) on the x-axis and position (x) on the y-axis.
2. Label the x-axis as "Time (s)" and the y-axis as "Position (m)".
3. Choose an appropriate scale for the axes based on the data values provided (e.g., increments of 0.5 for time and 50 for position).
4. Plot the given data points on the graph using the corresponding time and position values. Each point should be represented as (t,x).
(b) Determining the Velocity using Least-Squares Regression:
1. Calculate the average time (t̅) and average position (x̅) by summing up all the values and dividing by the number of data points.
- t̅ = (0.5 + 1.4 + 2.9 + 3.7 + 4.1) / 5
- x̅ = (39.9 + 75.4 + 139.3 + 186.5 + 193.2) / 5
2. Calculate the differences in time from the average time (Δt = t - t̅) and the differences in position from the average position (Δx = x - x̅) for each data point.
3. Calculate the sum of the products of Δt and Δx (Σ(Δt * Δx)) and the sum of the squares of Δt (Σ(Δt^2)).
4. Calculate the slope of the line (velocity) using the formula:
- velocity = Σ(Δt * Δx) / Σ(Δt^2)
5. Round the calculated velocity to three decimal places to get the final answer.
By following these steps, you can plot the data and determine the velocity of the car using the least-squares regression method.