Question

\[ \begin{aligned} \delta A_{\mathrm{tot}} & =\sqrt{[\delta(a b)]^{2}+[\delta(c d)]^{2}} \\ & =\sqrt{\left(0.095028 \mathrm{~m}^{2}\right)^{2}+\left(0.060896 \mathrm{~m}^{2}\right)^{2}} \\ & =0.112866 \mathrm{~m}^{2} \approx 0.1 \mathrm{~m}^{2} \end{aligned} \] We round the total uncertainty to one significant figure and round the best estimate to the same decimal place as the uncertainty. The total area with uncertainty is \[ A_{\mathrm{tot}}=(6.4 \pm 0.1) \mathrm{m}^{2} \] 2.2 Exercises Complete these calculations of uncertainty propagation and submit them for the pre-lab assignment. You must show your work in your submission. Your video analysis of the motion of a marble gives it position in frame 23 as \( \left(x_{23}, y_{23}\right)= \) \( (0.134 \mathrm{~m}, 0.120 \mathrm{~m}) \) and its position in frame 24 as \( \left(x_{24}, y_{24}\right)=(0.122 \mathrm{~m}, 0.112 \mathrm{~m}) \). You estimate that you can measure the \( x \) and \( y \) positions with uncertainty \( \pm 0.003 \mathrm{~m} \). The frame rate of the video is 30 frames/s, which means the time interval between frames is \( \Delta t=0.033333 \mathrm{~s} \). The uncertainty of the frame rate of a video camera is VERY small. For the sake of this problem, use \( \delta(\Delta t)=1 \times 10^{-6} \mathrm{~s} \). The mass of the marble is \( (2.031 \pm 0.001) \times 10^{-2} \mathrm{~kg} \). Calculate the following quantities: 1. The value of the velocity component \( v_{x}=\frac{x_{i+1}-x_{i}}{\Delta t} \) and its uncertainty, 2. The value of the velocity component \( v_{y}=\frac{y_{i+1}-y_{i}}{\Delta t} \) and its uncertainty, 3. The value of the velocity magnitude \( v=\sqrt{v_{x}^{2}+v_{y}^{2}} \) and its uncertainty, 4. The value of the momentum component \( p_{x}=m v_{x} \) and its uncertainty, 5. The value of the momentum component \( p_{y}=m v_{y} \) and its uncertainty, and 6. The value of the kinetic energy \( K=\frac{1}{2} m v^{2}=\frac{1}{2} m\left(v_{x}^{2}+v_{y}^{2}\right) \) and its uncertainty. Before you submit Write up your solutions to exercises 1-6 above clearly showing your work. Be sure to express your final values with uncertainties using the guidelines in Appendix A. Circle your answers. Make a PDF file with your work and submit it on Canvas in the Investigation 3 Pre-Lab assignment.

          \[
\begin{aligned}
\delta A_{\mathrm{tot}} & =\sqrt{[\delta(a b)]^{2}+[\delta(c d)]^{2}} \\
& =\sqrt{\left(0.095028 \mathrm{~m}^{2}\right)^{2}+\left(0.060896 \mathrm{~m}^{2}\right)^{2}} \\
& =0.112866 \mathrm{~m}^{2} \approx 0.1 \mathrm{~m}^{2}
\end{aligned}
\]
We round the total uncertainty to one significant figure and round the best estimate to the same decimal place as the uncertainty. The total area with uncertainty is
\[
A_{\mathrm{tot}}=(6.4 \pm 0.1) \mathrm{m}^{2}
\]
2.2 Exercises
Complete these calculations of uncertainty propagation and submit them for the pre-lab assignment. You must show your work in your submission.

Your video analysis of the motion of a marble gives it position in frame 23 as \( \left(x_{23}, y_{23}\right)= \) \( (0.134 \mathrm{~m}, 0.120 \mathrm{~m}) \) and its position in frame 24 as \( \left(x_{24}, y_{24}\right)=(0.122 \mathrm{~m}, 0.112 \mathrm{~m}) \). You estimate that you can measure the \( x \) and \( y \) positions with uncertainty \( \pm 0.003 \mathrm{~m} \). The frame rate of the video is 30 frames/s, which means the time interval between frames is \( \Delta t=0.033333 \mathrm{~s} \). The uncertainty of the frame rate of a video camera is VERY small. For the sake of this problem, use \( \delta(\Delta t)=1 \times 10^{-6} \mathrm{~s} \). The mass of the marble is \( (2.031 \pm 0.001) \times 10^{-2} \mathrm{~kg} \). Calculate the following quantities:
1. The value of the velocity component \( v_{x}=\frac{x_{i+1}-x_{i}}{\Delta t} \) and its uncertainty,
2. The value of the velocity component \( v_{y}=\frac{y_{i+1}-y_{i}}{\Delta t} \) and its uncertainty,
3. The value of the velocity magnitude \( v=\sqrt{v_{x}^{2}+v_{y}^{2}} \) and its uncertainty,
4. The value of the momentum component \( p_{x}=m v_{x} \) and its uncertainty,
5. The value of the momentum component \( p_{y}=m v_{y} \) and its uncertainty, and
6. The value of the kinetic energy \( K=\frac{1}{2} m v^{2}=\frac{1}{2} m\left(v_{x}^{2}+v_{y}^{2}\right) \) and its uncertainty.
Before you submit
Write up your solutions to exercises 1-6 above clearly showing your work. Be sure to express your final values with uncertainties using the guidelines in Appendix A. Circle your answers. Make a PDF file with your work and submit it on Canvas in the Investigation 3 Pre-Lab assignment.

δ Atot =√([δ(a b)]^2+[δ(c d)]^2)
=√((0.095028 m^2)^2+(0.060896 m^2)^2)
=0.112866 m^2≈ 0.1 m^2

We round the total uncertainty to one significant figure and round the best estimate to the same decimal place as the uncertainty. The total area with uncertainty is

Atot=(6.4 ± 0.1) m^2

2.2 Exercises
Complete these calculations of uncertainty propagation and submit them for the pre-lab assignment. You must show your work in your submission.

Your video analysis of the motion of a marble gives it position in frame 23 as (x23, y23)= (0.134 m, 0.120 m) and its position in frame 24 as (x24, y24)=(0.122 m, 0.112 m). You estimate that you can measure the x and y positions with uncertainty ± 0.003 m. The frame rate of the video is 30 frames/s, which means the time interval between frames is Δ t=0.033333 s. The uncertainty of the frame rate of a video camera is VERY small. For the sake of this problem, use δ(Δ t)=1 × 10^-6 s. The mass of the marble is (2.031 ± 0.001) × 10^-2 kg. Calculate the following quantities:
1. The value of the velocity component vx=(xi+1-xi)/(Δ t) and its uncertainty,
2. The value of the velocity component vy=(yi+1-yi)/(Δ t) and its uncertainty,
3. The value of the velocity magnitude v=√(vx^2+vy^2) and its uncertainty,
4. The value of the momentum component px=m vx and its uncertainty,
5. The value of the momentum component py=m vy and its uncertainty, and
6. The value of the kinetic energy K=(1)/(2) m v^2=(1)/(2) m(vx^2+vy^2) and its uncertainty.
Before you submit
Write up your solutions to exercises 1-6 above clearly showing your work. Be sure to express your final values with uncertainties using the guidelines in Appendix A. Circle your answers. Make a PDF file with your work and submit it on Canvas in the Investigation 3 Pre-Lab assignment.

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