Question

two vehicles start out traveling side by side along a straight road. their position functions, shown in the following graph, are given by s = f(t) and s = g(t), where s is measured in meters and t is measured in seconds. which vehicle has traveled farther at t = 2 seconds? vehicle f vehicle g what is the approximate velocity of the f vehicle at t = 3 seconds? meters per second what is the approximate velocity of the g vehicle at t = 3 seconds? meters per second which vehicle is traveling faster at t = 4 seconds? vehicle f vehicle g

Explanation:

Step1: Find positions at t = 2

From the graph, for vehicle \(f\), \(s=f(2)\approx 4\) meters and for vehicle \(g\), \(s = g(2)\approx 3\) meters. Since \(4>3\), vehicle \(f\) has traveled farther at \(t = 2\) seconds.

Step2: Approximate velocity of \(f\) at \(t=3\)

The velocity \(v\) of an object is the slope of the position - time graph. For vehicle \(f\) at \(t = 3\), we consider a small interval around \(t = 3\). Let's take the interval \([2,4]\). The change in position \(\Delta s=f(4)-f(2)\approx8 - 4=4\) meters and the change in time \(\Delta t=4 - 2 = 2\) seconds. The average velocity in this interval is \(v=\frac{\Delta s}{\Delta t}=\frac{4}{2}=2\) meters per second.

Step3: Approximate velocity of \(g\) at \(t = 3\)

For vehicle \(g\) at \(t = 3\), consider the interval \([2,4]\). The change in position \(\Delta s=g(4)-g(2)\approx7 - 3 = 4\) meters and the change in time \(\Delta t=4 - 2=2\) seconds. The average velocity in this interval is \(v=\frac{\Delta s}{\Delta t}=\frac{4}{2}=2\) meters per second.

Step4: Determine faster - moving vehicle at \(t = 4\)

The slope of the position - time graph represents the velocity. At \(t = 4\), the slope of the graph of \(f\) is steeper than the slope of the graph of \(g\). So vehicle \(f\) is traveling faster at \(t = 4\) seconds.

Answer:

1. Vehicle f
2. 2
3. 2
4. Vehicle f