To determine the relationship between force and acceleration from the given data, we can perform a linear regression analysis. This involves finding the best-fit line that describes the relationship between the two variables.
Given Data Table:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Force (N)} & \text{Acceleration (m/s}^2\text{)} \\
\hline 0 & 0 \\
\hline 150 & 1 \\
\hline 200 & 1.3 \\
\hline 250 & 1.6 \\
\hline 350 & 2.3 \\
\hline 400 & 2.6 \\
\hline
\end{array}
\][/tex]
1. Collect Data Points:
[tex]\[
(0, 0), (150, 1), (200, 1.3), (250, 1.6), (350, 2.3), (400, 2.6)
\][/tex]
2. Perform Linear Regression:
We aim to fit a line [tex]\( y = mx + b \)[/tex] where [tex]\( y \)[/tex] is the acceleration, and [tex]\( x \)[/tex] is the force. Here, [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Compute the Slope (m) and Intercept (b):
After performing the calculations, we find:
- Slope ([tex]\( m \)[/tex]): [tex]\( 0.006506024096385543 \)[/tex]
- Intercept ([tex]\( b \)[/tex]): [tex]\( 0.0028112449799198133 \)[/tex]
4. Interpret the Slope (m):
- The positive slope [tex]\( m \)[/tex] indicates that as the force increases, the acceleration also increases.
- Since [tex]\( m \)[/tex] is positive and significant, it shows a positive linear relationship between force and acceleration.
5. Determine the Relationship:
The positive slope confirms that force is directly proportional to acceleration. According to Newton's Second Law of Motion ([tex]\( F = ma \)[/tex]), for a constant mass [tex]\( m \)[/tex], the force [tex]\( F \)[/tex] is directly proportional to the acceleration [tex]\( a \)[/tex].
Thus, based on the computations and derived relationship:
- Force is directly proportional to acceleration.
