91影视

The graph of $\mathrm{a}(\mathrm{m}/{\mathrm{s}}^2)$ against $\mathrm{t}(\mathrm{ms})$ for bare head and helmet is plotted. The scaling on the vertical axis is set by ${\mathrm{a}}_{\mathrm{s}}=200鈥�\mathrm{m}/{\mathrm{s}}^2$.

The calculation of the region bounded by the curve within the given graph is called the area under the curve.The velocity can be determined by the graphical integration of acceleration versus time.

Divide the curve into 4 different parts:

From 0 to 2 ms, has the shape of a triangle, with the area

$$\begin{gathered} {\mathrm{Area}}_{0-2}=\frac{1}{2}脳(2脳{10}^{-3})脳120 \\ =0.12\mathrm{m}/\mathrm{s} \end{gathered}$$

From 2 ms to 4 ms , has the shape of trapezoidal, with the area

$$\begin{gathered} {\mathrm{Area}}_{2-4}=\frac{1}{2}脳\mathrm{base}脳(\mathrm{sum}\mathrm{of}\mathrm{parallel}\mathrm{sides}) \\ =\frac{1}{2}脳(2脳{10}^{-3})脳(120+140) \\ =\frac{1}{2}脳(2脳{10}^{-3})脳260 \\ =0.26\mathrm{m}/\mathrm{s} \end{gathered}$$

From 4 to 6 ms , has the shape of trapezoidal, with area

localid="1656305016065" $$\begin{gathered} {\mathrm{Area}}_{4-6}=\frac{1}{2}脳\mathrm{base}脳(\mathrm{sum}\mathrm{of}\mathrm{parallel}\mathrm{sides}) \\ =\frac{1}{2}脳(2脳{10}^{-3})脳(140+200) \\ =\frac{1}{2}脳(2脳{10}^{-3})脳340 \\ =0.34\mathrm{m}/\mathrm{s} \end{gathered}$$

From 6 to 7 ms, has the shape of triangle, with area

$$\begin{gathered} {\mathrm{Area}}_{6-7}=\frac{1}{2}脳B\mathrm{ase}脳Height \\ =\frac{1}{2}脳(1脳{10}^{-3})脳200 \\ =0.10\mathrm{m}/\mathrm{s} \end{gathered}$$

From 0 to 3 ms , has the shape of triangle, with area

$$\begin{gathered} {\mathrm{Area}}_{0-3}=\frac{1}{2}脳\mathrm{base}脳\mathrm{Height} \\ =\frac{1}{2}脳(3脳{10}^{-3})脳40 \\ =0.060\mathrm{m}/\mathrm{s} \end{gathered}$$

From 3 to 4 , has the shape of rectangle, with area

$$\begin{gathered} {\mathrm{Area}}_{3-4}=\mathrm{base}脳\mathrm{height} \\ =(1脳{10}^{-3})脳40 \\ =0.040\mathrm{m}/\mathrm{s} \end{gathered}$$

From 4 to 6 , has the shape of the trapezoidal, with area

$$\begin{gathered} {\mathrm{Area}}_{4-6}=\frac{1}{2}脳\mathrm{base}脳\left(\mathrm{sum}\mathrm{of}\mathrm{parallel}\mathrm{sides}\right) \\ =\frac{1}{2}脳(2脳{10}^{-3})脳(40+80) \\ =\frac{1}{2}脳(2脳{10}^{-3})脳120 \\ =0.12\mathrm{m}/\mathrm{s} \end{gathered}$$

From 6 to 7 ms , has the shape of triangle, with area

$$\begin{gathered} {\mathrm{Area}}_{6-7}=\frac{1}{2}脳\mathrm{Base}脳\mathrm{Height} \\ =\frac{1}{2}脳(1脳{10}^{-3})脳80 \\ =0.040\mathrm{m}/\mathrm{s} \end{gathered}$$

Total area under the bare curve will be,

$$\begin{gathered} {\mathrm{V}}_{\mathrm{Bare}}={\mathrm{Area}}_{0-2}+{\mathrm{Area}}_{2-4}+{\mathrm{Area}}_{4-6}+{\mathrm{Area}}_{6-7} \\ =0.12+0.26+0.34+0.10 \\ =0.82\mathrm{m}/\mathrm{s} \end{gathered}$$

Total area under the helmeted curve will be,

$$\begin{gathered} {\mathrm{V}}_{\mathrm{helm}\mathrm{eted}}={\mathrm{Area}}_{0-3}+{\mathrm{Area}}_{3-4}+{\mathrm{Area}}_{4-6}+{\mathrm{Area}}_{6-7} \\ =0.060+0.040+0.12+0.040 \\ =0.26\mathrm{m}/\mathrm{s} \end{gathered}$$

The difference between the two curves will be,

$$\begin{gathered} 鈭�\mathrm{V}={\mathrm{V}}_{\mathrm{Bare}}-{\mathrm{V}}_{\mathrm{helmeted}} \\ =0.82-0.26 \\ =0.56\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the difference between the speeds will be 0.56 m/s .