A race car moves such that its position fits the relationship
x = (5.0 m/s)t + (0.75 m/s3)t3
where x is measured in meters and t in seconds. (a) Plot a graph of the car s position versus time. (b) Determine the instantaneous velocity of the car at t = 4.0 s, using time intervals of 0.40 s, 0.20 s, and 0.10 s. (c) Compare the average velocity during the first 4.0 s with the results of (b).
I have already answered this question. See my answer posted earlier.
b) Here's the answer using differential calculus:
Instantaneous velocity = dx/dt = 5.0 m/s + (3)(0.75)(t^2);
Evaluated at 4.0 seconds:
Instantaneous velocity = 5.0 + (2.25)(16) = 41 m/s
To plot a graph of the car's position versus time, we can use the given relationship:
x = (5.0 m/s)t + (0.75 m/s^3)t^3
To graph this relationship, we need to assign values to t and calculate the corresponding values of x. Let's choose a range of values for t to plot.
(a) Plotting the Position vs. Time Graph:
Let's choose a time range from t = 0 to t = 4 seconds and use intervals of 0.1 seconds. We can calculate the position for each value of t using the given relationship.
For example,
At t = 0 seconds, x = (5.0 m/s)(0 s) + (0.75 m/s^3)(0 s^3) = 0 meters.
At t = 0.1 seconds, x = (5.0 m/s)(0.1 s) + (0.75 m/s^3)(0.1 s^3) = 0.05 meters.
By calculating x for each chosen value of t within the time range, we can create data points and plot them on a graph with x on the y-axis and t on the x-axis.
(b) Determining Instantaneous Velocities at t = 4.0 s:
To find the instantaneous velocity at t = 4.0 s, we need to calculate the slope of the position vs. time graph at that particular point. We can do this by calculating the average velocity over smaller time intervals near t = 4.0 s.
Using the given time intervals of 0.40 s, 0.20 s, and 0.10 s, we can determine the average velocities for each interval.
For example, for t = 4.0 s and a time interval of 0.40 s:
Velocity = (Change in position) / (Time interval)
= (x2 - x1) / (t2 - t1)
= (x(t + 0.40 s) - x(t)) / (0.40 s)
By calculating this for each given time interval, we can determine the instantaneous velocity at t = 4.0 s.
(c) Comparing Average Velocity During the First 4.0 s:
To compare the average velocity during the first 4.0 s with the results of part (b), we need to find the average velocity over the entire time interval from t = 0 to t = 4.0 s.
Average Velocity = (Change in position) / (Total time)
= (x(4.0 s) - x(0)) / (4.0 s)
By calculating this average velocity, we can compare it with the instantaneous velocities obtained in part (b) for different time intervals.