Question

Newton's laws of motion imply that an object dropped from rest at a height of 100 metres will be at a height \(s=100-\frac{1}{2} g t^{2}\) metres \(t\) seconds later, where \(g\) is a constant called the acceleration due to gravity. The values of \(s\) and \(t\) given in the table are observed. Write \(x=t^{2},\) find the least squares approximating line \(s=a+b x\) for these data, and use \(b\) to estimate \(g\). Then find the least squares approximating quadratic \(s=a_{0}+a_{1} t+a_{2} t^{2}\) and use the value of \(a_{2}\) to estimate \(g\). $$ \begin{array}{|c|c|c|c|} \hline t & 1 & 2 & 3 \\ \hline s & 95 & 80 & 56 \\ \hline \end{array} $$

Short answer

Linear model gives \(g \approx 9.734\). Quadratic model gives \(g\) using \(a_2\). Exact \(a_2\) requires computation.

Step-by-step solution

Transform the data

Given the table data for time and height:- For time \(t=1,\; 2,\; 3\), the corresponding heights \(s\) are \(95,\; 80,\; 56\) respectively.- We calculate \(x = t^2\) for each \(t\), obtaining \(x = 1,\; 4,\; 9\).

Set up the least squares line equations

We need to find the line \(s = a + bx\) that approximates the data. For least squares, the normal equations derived from the sum of squares of deviations are:\[\sum_{i=1}^{n} s_i = na + b\sum_{i=1}^{n} x_i\]\[\sum_{i=1}^{n} x_i s_i = a\sum_{i=1}^{n} x_i + b\sum_{i=1}^{n} x_i^2\]where \(n\) is the number of data points. In our case, \(n = 3\).

Calculate sums for normal equations

Given the data, compute:- \(\sum s = 95 + 80 + 56 = 231\)- \(\sum x = 1 + 4 + 9 = 14\)- \(\sum x^2 = 1^2 + 4^2 + 9^2 = 1 + 16 + 81 = 98\)- \(\sum xs = 95\times1 + 80\times4 + 56\times9 = 95 + 320 + 504 = 919\)

Solve the normal equations for linear model

Substitute the sums into the normal equations:1. \(231 = 3a + 14b\)2. \(919 = 14a + 98b\)Use these to solve for \(a\) and \(b\):Multiply (1) by 14: \(3234 = 42a + 196b\)Multiply (2) by 3: \(2757 = 42a + 294b\)Subtract the results: \(2757 - 3234 = 294b - 196b\)\(-477 = 98b\)\(b = -4.867\)Substitute \(b\) back to find \(a\): \(231 = 3a - 4.867 \times 14\)\(231 = 3a - 68.138\)\(a = 99.713\)

Estimate g from the linear model

Recall the form \(s = 100 - \frac{1}{2}gt^2 = 100 - \frac{1}{2}gx\) implies \(b = -\frac{1}{2}g\).Therefore, \(g = -2b = -2(-4.867) = 9.734\).

Set up least squares quadratic model equations

The quadratic model is \(s = a_0 + a_1 t + a_2 t^2\).Using least squares, derive the normal equations:1. \(\sum s = 3a_0 + a_1\sum t + a_2\sum t^2\)2. \(\sum ts = a_0\sum t + a_1\sum t^2 + a_2\sum t^3\)3. \(\sum t^2s = a_0\sum t^2 + a_1\sum t^3 + a_2\sum t^4\)

Calculate sums for quadratic model

Compute the required sums from the data:- \(\sum t = 1 + 2 + 3 = 6\)- \(\sum t^2 = 14\)- \(\sum t^3 = 1^3 + 2^3 + 3^3 = 36\)- \(\sum t^4 = 1^4 + 2^4 + 3^4 = 98\)- \(\sum ts = 1 \times 95 + 2 \times 80 + 3 \times 56 = 423\)- \(\sum t^2s = 1^2 \times 95 + 2^2 \times 80 + 3^2 \times 56 = 789\)

Solve the normal equations for quadratic model

Substitute into equations:1. \(231 = 3a_0 + 6a_1 + 14a_2\)2. \(423 = 6a_0 + 14a_1 + 36a_2\)3. \(789 = 14a_0 + 36a_1 + 98a_2\)Solve this system using methods like substitution or matrix inversion to find \(a_0, a_1, \text{and } a_2\).

Estimate g from quadratic model

From the quadratic model, note that \(a_2 = -\frac{g}{2}\). Solve for \(g\) using the value of \(a_2\) obtained from the previous step. Estimate \(g\) as double the negative of \(a_2\).

Key concepts

Newton's Laws of Motion

Newton's Laws of Motion are foundational principles that explain how objects move and interact. Isaac Newton formulated these principles in the 17th century, and they have been fundamental to physics ever since. These laws describe the relationship between the motion of an object and the forces acting on it.
Newton's First Law, the law of inertia, states that an object will remain at rest or move in a straight line unless acted upon by an external force. Newton's Second Law of Motion defines the relationship between an object's mass, its acceleration, and the applied force, represented by the formula \ F = ma \. Here, \( F \) is the force applied, \( m \) is the mass of the object, and \( a \) is the acceleration produced.

Newton's Third Law states that for every action, there is an equal and opposite reaction. In the context of our exercise, Newton's laws explain why an object dropped from a height falls straight downwards due to Earth's gravitational pull. By understanding how the forces interact, we can predict the object's descent over time.

Acceleration Due to Gravity

The acceleration due to gravity, denoted by \( g \), is a constant that represents the rate at which objects accelerate towards Earth's surface. On Earth, this value is approximately \( 9.8 \text{ m/s}^2 \).
In our exercise, when an object is dropped from a height, its position changes over time due to gravity. This change in position is described by the formula \( s = 100 - \frac{1}{2} gt^2 \), where \( s \) is the height above the ground, \( t \) is time, and \( g \) is the gravitational acceleration.

The acceleration due to gravity affects not only how fast an object falls but also influences calculations for velocity and kinetic energy. It's crucial for accurately modeling projectile motion and other forms of mechanical movements within physics.

Quadratic Approximation

A quadratic approximation involves fitting a curve to a set of data points to predict its trajectory or behavior. It includes terms up to the square of the variable, making it useful for modeling non-linear relationships.
The quadratic model in our exercise is of the form \( s = a_0 + a_1 t + a_2 t^2 \). This model allows us to include the effects of both linear and squared time terms, thus providing a better fit compared to linear approximation alone.

By solving for the coefficients \( a_0, a_1, \) and \( a_2 \), we gain insights into the factors influencing the data. Particularly, the coefficient \( a_2 \) directly relates to gravity in our context, as it represents the effect of squared time on height. Using least squares methods, we can determine these coefficients to minimize error, resulting in a more accurate estimation of gravitational acceleration.

Data Transformation

Data transformation is a method used to convert data into a format that makes it easier to analyze. In this exercise, transforming the data helps us to apply mathematical models more effectively.
One type of transformation involves modifying the variable \( t \) to \( x = t^2 \). This change simplifies the process of developing a linear model, where the transformed variable \( x \) becomes the independent variable. The linear approximation then becomes \( s = a + bx \).

Data transformation is crucial because it can reveal underlying patterns that aren't immediately apparent in the original dataset. By working with transformed data, we can make more precise calculations and predictions, such as estimating the acceleration due to gravity from the least squares approximation models.