Example from Physics, 2.4 Velocity vs. Time Graphs

Using Curvy Velocity Graph to Calculate Some Stuff: Jet Car, Take Two

Use Figure 2.19 to (a) find the approximate displacement of the jet car over the time shown, (b) calculate the instantaneous acceleration at t = 30 s, (c) find the instantaneous velocity at 30 s, and (d) calculate the approximate average velocity over the interval shown.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Strategy
1. Because this graph is an undefined curve, we have to estimate shapes over smaller intervals in order to find the areas.
2. Like when we were working with a curved displacement graph, we will need to take a tangent line at the instant we are interested and use that to calculate the instantaneous acceleration.
3. The instantaneous velocity can still be read off of the graph.
4. We will find the average velocity the same way we did in the previous example.
2
Solution
1. 1. This problem is more complicated than the last example. To get a good estimate, we should probably break the curve into four sections. 0 → 10 s, 10 → 20 s, 20 → 40 s, and 40 → 70 s.
  2. Calculate the bottom rectangle (common to all pieces). 165 m/s $\times$ 70 s = 11,550 m.
  3. Estimate a triangle at the top, and calculate the area for each section. Section 1 contributes a triangle of width 10 s and height (210 – 165) m/s = 45 m/s so the added areas for Section 1 = (1/2)(45 m/s)(10 s) = 225 m; section 2 contributes a rectangle and a triangle with area = 100 m + 450 m = 550 m; section 3 = 150 m + 1,300 m = 1,450 m; section 4 = 2,550 m.
  4. Add them together to get a net displacement of 16,325 m.
2. Using the tangent line given, we find that the slope is 1 m/s².
3. The instantaneous velocity at t = 30 s, is 240 m/s.
4. 1. Find the net displacement, which we found in part (a), was 16,325 m.
  2. Find the total time, which for this case is 70 s.
  3. Divide $\frac{16,325 m}{70 s} \sim 233 m/s$

0 of 2 steps open

How did it go?

Strategy

Solution

Strategy

Solution