Problem Breakdown:

We need to determine the time interval $\Delta t$ that allows a physics class to create a data table modeling the 1-D motion of a basketball falling from rest, with constant acceleration. The error in the total distance traveled by the ball should not exceed 4%.

Key Information:

• Acceleration $a = 9.8 \, \text{m/s}^2$
• Total time $t = 3.1 \, \text{seconds}$
• Error in total distance $= 4\%$
• The equations provided:
  • $v_{i+1} = v_i + a \Delta t$
  • $x_{i+1} = x_i + v_i \Delta t$

Step 1: Exact Distance Traveled

The exact distance traveled by the ball can be found using the kinematic equation for distance under constant acceleration: $d = \frac{1}{2} a t^2$ Substituting the given values: $d = \frac{1}{2} \times 9.8 \, \text{m/s}^2 \times (3.1 \, \text{s})^2$ Let's calculate this distance.

The exact distance traveled by the basketball is $47.09 \, \text{meters}$.

Step 2: Error Requirement

The error in the total distance must not exceed 4%. Therefore, the allowed error in distance is: $\text{Allowed error} = 0.04 \times 47.09 = 1.8836 \, \text{meters}$ This means the calculated distance from the table using the discrete time steps should be within $1.88 \, \text{meters}$ of the exact value.

Step 3: Approximation with Time Intervals $\Delta t$

We will now estimate how the discrete steps for $\Delta t$ affect the distance calculation. Using smaller $\Delta t$, we can reduce the error. To find the suitable value of $\Delta t$, we need to run an iterative process, gradually decreasing $\Delta t$, and checking how close the result is to the exact distance.

Let's perform the simulation using the equations for $v_{i+1}$ and $x_{i+1}$, and check for the smallest $\Delta t$ that keeps the error within the allowed range.

It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through a step-by-step approach to manually find the time interval $\Delta t$ or provide further insights into the physics problem.

Would you like details on how to proceed, or do you have any specific questions about the process? Here are some relative questions to deepen your understanding:

1. How does increasing or decreasing $\Delta t$ affect the accuracy of distance estimation?
2. Can we use different methods to minimize error in numerical simulations?
3. How do we determine the velocity at any point during the ball's fall?
4. What impact would air resistance have if included in the model?
5. Why is the error tolerance important in numerical methods for physics?

Tip: When performing numerical simulations, smaller time steps generally lead to more accurate results, but at the cost of increased computational complexity.