To calculate the acceleration for frame 3, we need to determine the acceleration between frames 2 and 4. Acceleration can be calculated as the change in velocity over time. Here's how we can break it down:

Step 1: Calculate velocity at frame 2, 3, and 4

The velocity at each frame can be approximated by the change in position divided by the change in time between consecutive frames:

$\text{Velocity} = \frac{\Delta X}{\Delta t}$

Where $\Delta X$ is the change in position, and $\Delta t$ is the change in time.

1. Velocity between Frame 2 and Frame 3:

   • $\Delta X = X_3 - X_2 = 36.51 - 20.04 = 16.47 \, \text{m}$
   • $\Delta t = t_3 - t_2 = 3.28 - 2.11 = 1.17 \, \text{s}$

   $\text{Velocity}_{2 \rightarrow 3} = \frac{16.47}{1.17} = 14.06 \, \text{m/s}$

2. Velocity between Frame 3 and Frame 4:

   • $\Delta X = X_4 - X_3 = 56.4 - 36.51 = 19.89 \, \text{m}$
   • $\Delta t = t_4 - t_3 = 4.15 - 3.28 = 0.87 \, \text{s}$

   $\text{Velocity}_{3 \rightarrow 4} = \frac{19.89}{0.87} = 22.86 \, \text{m/s}$

Step 2: Calculate the acceleration

Now that we have the velocities at frames 2, 3, and 4, we can calculate the acceleration by finding the change in velocity between frames 2 and 4, divided by the change in time between these frames:

$\text{Acceleration} = \frac{\Delta v}{\Delta t}$

Where $\Delta v$ is the change in velocity, and $\Delta t$ is the change in time between frames 2 and 4.

• $\Delta v = \text{Velocity}_{3 \rightarrow 4} - \text{Velocity}_{2 \rightarrow 3} = 22.86 - 14.06 = 8.8 \, \text{m/s}$
• $\Delta t = t_4 - t_2 = 4.15 - 2.11 = 2.04 \, \text{s}$

$\text{Acceleration} = \frac{8.8}{2.04} \approx 4.31 \, \text{m/s}^2$

Conclusion:

The acceleration between frames 2 and 4 is approximately $4.31 \, \text{m/s}^2$.

Let me know if you would like further explanation or if you have other questions!

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Here are some follow-up questions to expand on the concept:

1. How would you calculate the instantaneous acceleration if you had more data points?
2. Why is it important to use velocity over small intervals when calculating acceleration?
3. How does the motion of the cheetah change with respect to acceleration in this case?
4. What would happen if you used a different method, like calculating average velocity over a larger interval?
5. How does this acceleration compare to the typical acceleration of a cheetah in real-life chases?

Tip: When dealing with raw data like this, always ensure that you have a consistent time scale and units across the measurements to avoid errors in calculation.