. 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 in 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 0.10s. (c) Compare the average velocity during the first 4.0 with the results of part (b).
To answer this question, let's break it down step by step:
(a) Plot a graph of the car's position versus time:
To plot the graph of the car's position versus time, we need to substitute different values of t into the equation x = (5.0 m/s)t + (0.75 m/s^3)t^3 and calculate the corresponding values of x. Let's consider various values of t ranging from 0 to a chosen final time, such as 10 seconds. For each value of t, calculate the corresponding value of x using the given equation. Then, plot the values of x on the y-axis against the corresponding values of t on the x-axis. Connect these points to get a smooth curve representing the position versus time graph.
(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:
To determine the instantaneous velocity at a specific time, we can calculate the derivative of the position function with respect to time and evaluate it at that specific time. In this case, the derivative of the position equation x(t) = (5.0 m/s)t + (0.75 m/s^3)t^3 would give us the expression for velocity.
To obtain the derivative, differentiate each term of the position equation separately. The derivative of (5.0 m/s)t with respect to t is just 5.0 m/s (as it's a linear term). Similarly, differentiating (0.75 m/s^3)t^3 with respect to t gives us (2.25 m/s^3)t^2.
So, the expression for velocity (v) as a function of time (t) is: v(t) = 5.0 m/s + (2.25 m/s^3)t^2.
To determine the instantaneous velocity at t = 4.0 s, substitute t = 4.0 s into the velocity equation and calculate the value.
Using the given time intervals (0.40 s, 0.20 s, and 0.10 s), you can also determine average velocities by taking the change in position over time intervals and dividing them: average velocity = (change in position) / (time interval).
(c) Compare the average velocity during the first 4.0 s with the results of part (b):
To compare the average velocity during the first 4.0 s with the results of part (b), calculate the average velocity over the interval from t = 0 to t = 4 seconds using the average velocity formula mentioned in part (b). Then, compare this value with the instantaneous velocity at t = 4.0 s calculated in part (b).