Question

1. what is the displacement of the object between 4 and 16 seconds? (a) zero (b) 32m (c) -32m (d) 64m (e) -64m 21. what is the average velocity of the object between 0 and 8 seconds? (a) zero (b) 8m/s (c) 1m/s (d) -8m/s (e) -1m/s

Explanation:

20. Displacement between 4 and 16 seconds

Step1: Recall the formula for displacement from a velocity - time graph

Displacement \(d=\int_{t_1}^{t_2}v(t)dt\), which is the area under the velocity - time graph between \(t = 4\ s\) and \(t=16\ s\).
The area of a triangle is \(A=\frac{1}{2}bh\) and the area of a rectangle is \(A = bh\).

Step2: Calculate the area of the triangle from \(t = 4\ s\) to \(t = 6\ s\)

The base of the triangle \(b_1=6 - 4=2\ s\), and the height \(h_1=- 4\ m/s\) (negative because it's below the time - axis). The area \(A_1=\frac{1}{2}\times2\times(-4)=- 4\ m\).

Step3: Calculate the area of the rectangle from \(t = 6\ s\) to \(t = 14\ s\)

The base \(b_2=14 - 6 = 8\ s\), and the height \(h_2=-4\ m/s\). The area \(A_2=8\times(-4)=-32\ m\).

Step4: Calculate the area of the triangle from \(t = 14\ s\) to \(t = 16\ s\)

The base \(b_3=16 - 14=2\ s\), and the height \(h_3 = 0\ m/s\) (starts from \(v=-4\ m/s\) and goes to \(v = 0\ m/s\)). The area \(A_3=\frac{1}{2}\times2\times0=0\ m\).

Step5: Sum up the areas

\(d=A_1 + A_2+A_3=-4-32 + 0=-32\ m\).

21. Average velocity between 0 and 8 seconds

Step1: Recall the formula for average velocity

Average velocity \(\bar{v}=\frac{\int_{0}^{8}v(t)dt}{8 - 0}\), where \(\int_{0}^{8}v(t)dt\) is the area under the velocity - time graph from \(t = 0\ s\) to \(t = 8\ s\).

Step2: Calculate the area of the triangle from \(t = 0\ s\) to \(t = 6\ s\)

The base \(b_1=6\ s\), and the height \(h_1 = 4\ m/s\). The area \(A_1=\frac{1}{2}\times6\times4 = 12\ m\).

Step3: Calculate the area of the triangle from \(t = 6\ s\) to \(t = 8\ s\)

The base \(b_2=8 - 6=2\ s\), and the height \(h_2=-4\ m/s\). The area \(A_2=\frac{1}{2}\times2\times(-4)=-4\ m\).

Step4: Sum up the areas

\(\int_{0}^{8}v(t)dt=A_1+A_2=12-4 = 8\ m\).

Step5: Calculate the average velocity

\(\bar{v}=\frac{8}{8}=1\ m/s\).

Answer:

1. C. - 32m
2. C. 1m/s