A ball moves along a 1D coordinate system with the following motion parameters at time t = 0 s:  x0 = 6.7 m and v0 = 0 m/s.  The ball has a constant acceleration of 2.6 m/s2 during the entire motion which occurs over a time of 3.8 s.  Assume that an error in position of 2 % is acceptable in this application over the entire motion.  How many rows N will be needed in the table to model the entire motion to within the required error even though you are leaving out the final term of 1/2 a (Δt)2 ) present in the exact position expression?  

The ball's motion is defined by the following known parameters:

- Initial position: \( x_0 = 6.7 \, \text{m} \)
- Initial velocity: \( v_0 = 0 \, \text{m/s} \)
- Constant acceleration: \( a = 2.6 \, \text{m/s}^2 \)
- Time of motion: \( t = 3.8 \, \text{s} \)
- Acceptable position error: 2% of the exact position.

### Exact Position Equation

The exact position of the ball at any time \( t \) is given by the equation of motion:
\[
x(t) = x_0 + v_0 t + \frac{1}{2} a t^2
\]
Since \( v_0 = 0 \), this simplifies to:
\[
x(t) = x_0 + \frac{1}{2} a t^2
\]

For \( t = 3.8 \, \text{s} \), the exact position is:
\[
x(3.8) = 6.7 + \frac{1}{2} (2.6) (3.8)^2
\]

### Approximate Position Model Without the Final Term

The position model without the final term (i.e., \( \frac{1}{2} a (\Delta t)^2 \)) would only take into account uniform motion (constant velocity), assuming a table row-wise update. 

You need to determine how many rows \( N \) are required to model this motion such that the difference between the exact and approximate position is within a 2% acceptable error.

Let me calculate this step-by-step.The exact position of the ball at \( t = 3.8 \, \text{s} \) is \( 25.47 \, \text{m} \), and the acceptable error in position is \( 0.5094 \, \text{m} \).

Next, we will determine how many table rows \( N \) are needed to ensure the position error is within this limit by leaving out the final term \( \frac{1}{2} a (\Delta t)^2 \) in the approximate position model. This requires estimating the maximum allowable time step \( \Delta t \). Let's continue.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can explain how to proceed with the calculations step-by-step manually or guide you through the method!

Would you like further details or have any questions? Here are some related topics that may help:

1. How to approximate errors in numerical models.
2. Calculating motion with constant acceleration.
3. Estimating time steps in physics simulations.
4. How truncation affects the accuracy of physical models.
5. Kinematic equations and their applications.

**Tip:** Always assess the acceptable error when approximating results in physical models to ensure accuracy within required limits.

A ball moves along a 1D coordinate system with the following motion parameters at time t = 0 s: x0 = 6.7 m and v0 = 0 m/s. The ball has a constant acceleration of 2.6 m/s² during the entire motion which occurs over a time of 3.8 s. Assume that an error in position of 2 % is acceptable in this application over the entire motion. How many rows N will be needed in the table to model the entire motion to within the required error even though you are leaving out the final term of 1/2 a (Δt)² present in the exact position expression?

Solve a kinematics problem involving a ball moving under constant acceleration (2.6 m/s²) and calculate the number of rows needed to model the motion within a 2% position error. Learn how to apply kinematic equations and estimate errors in numerical models.

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