Problem 99 - Physics for Scientists and Engineers with Modern Physics

Question

Students shoot a plastic ball horizontally from a projectile launcher. They measure the distance \(x\) the ball travels horizontally, the distance \(y\) the ball falls vertically, and the total time \(t\) the ball is in the air for six different heights of the projectile launcher. Here is their data. $$ \begin{array}{ccc} \hline \text { Time, } & \text { Horizontal distance, } & \text { Vertical distance, } \\ t(\mathrm{~s}) & x(\mathrm{~m}) & y(\mathrm{~m}) \\ \hline 0.217 & 0.642 & 0.260 \\ 0.376 & 1.115 & 0.685 \\ 0.398 & 1.140 & 0.800 \\ 0.431 & 1.300 & 0.915 \\ 0.478 & 1.420 & 1.150 \\ 0.491 & 1.480 & 1.200 \\ \hline \end{array} $$ (a) Determine the best-fit straight line that represents \(x\) as a function of \(t .\) What is the initial speed of the ball obtained from the best-fit straight line? ( \(b\) ) Determine the best-fit quadratic equation that represents \(y\) as a function of \(t .\) What is the acceleration of the ball in the vertical direction?

Step-by-Step Solution

Answer

Initial speed is the slope from the linear regression of x vs t. The acceleration is twice the coefficient of t² from the y vs t quadratic regression.

Step 1: Plot Horizontal Distance vs. Time

Take the data for time (\(t\)) and horizontal distance (\(x\)) and plot these points on a graph. The graph should have time on the x-axis and horizontal distance on the y-axis. This visualization will help us determine the best-fit straight line.

Step 2: Determine Best-fit Line for x vs. t

Using a linear regression analysis or a graphing calculator, find the equation of the best-fit line that represents the relationship between horizontal distance \(x\) and time \(t\). This line will be in the form \(x = vt + x_0\), where \(v\) is the slope representing the velocity, and \(x_0\) is the intercept.

Step 3: Calculate Initial Speed

The slope of the best-fit line, determined in the previous step, represents the initial horizontal speed of the ball. Use the equation of the line to identify this slope.

Step 4: Plot Vertical Distance vs. Time

Now take the data for time (\(t\)) and vertical distance (\(y\)) and plot these points on another graph. Again, time should be on the x-axis and vertical distance on the y-axis.

Step 5: Determine Best-fit Quadratic Equation for y vs. t

Using quadratic regression analysis or a graphing tool, find the best-fit quadratic equation that represents the relationship between \(y\) (vertical distance) and \(t\) (time). This equation will be in the form \(y = at^2 + bt + c\), where \(a\) represents half the vertical acceleration \(g\).

Step 6: Calculate Vertical Acceleration

From the quadratic equation obtained, identify the coefficient \(a\). Since the equation is \(y = \frac{1}{2}gt^2\) for free fall under gravity, \(2a\) gives the acceleration due to gravity \(g\). Thus, multiply \(a\) by 2 to obtain the vertical acceleration.

Key Concepts

Horizontal motion

When discussing projectile motion, understanding horizontal motion is key. When an object moves horizontally, it covers a distance known as horizontal distance or displacement. This happens without any influence from gravity over the horizontal trajectory.
In the given exercise, horizontal motion involves measuring how far the plastic ball travels horizontally after being launched. Since there is no air resistance considered, the horizontal motion follows a constant speed, and this means it doesn’t accelerate or decelerate in the horizontal plane.
To analyze horizontal motion, we plot the horizontal distance against time. This straightforward analysis allows us to model how the ball behaves as it moves across the horizontal distance by determining a best-fit line. The linear relationship from this plot reveals the ball's velocity, which remains constant in horizontal motion without any extra horizontal forces.

Vertical motion

Vertical motion is slightly more complex than horizontal motion due to gravity. In vertical motion, the projectile’s path is influenced by gravitational force pulling the object towards the Earth. When these students shoot the ball, it not only moves horizontally but also falls downward due to gravity.
This downward fall, or vertical motion, can be observed by measuring how far the ball drops vertically over time, as in the data collected for this exercise. Unlike horizontal motion, the vertical distance covered (or vertical displacement) changes due to the constant acceleration caused by gravity.
Evaluating this vertical motion involves plotting vertical distance against time and finding an appropriate curve, often a parabola, which represents this change over time. The best-fit quadratic equation gives insight into this downward path, allowing us to quantify gravity's influence on the ball's vertical travel.

Initial speed

Initial speed, or initial velocity, is an essential aspect of projectile motion analysis. It's the speed at which the ball leaves the launcher from rest. Initial speed influences both how far and how fast the ball travels in a given direction.
In horizontal motion, the initial speed is determined by the ball's horizontal velocity, which can be derived from the slope of the best-fit line of the horizontal distance versus time graph. In this straight-line relationship, the slope essentially equates to the initial horizontal velocity."
Knowing this initial speed enables predictions of the projectile’s future position and time of travel under similar conditions. Thus, for anyone studying motion, the initial speed provides an important reference point for both further calculations and understanding of the projectile’s behavior.

Quadratic equations

Quadratic equations provide a mathematical framework to describe the vertical motion of projectiles, especially when acceleration is involved, such as with gravity.
In the exercise, the quadratic equation derived from the vertical distance versus time graph represents how vertical displacement changes over a period. The equation typically follows the form \(y = at^2 + bt + c\), where \(y\) is the vertical distance, \(t\) the time, and \(a\), \(b\), and \(c\) are constants that describe the parabola's curvature.
This approach helps in visualizing how acceleration affects the ball's vertical movement. By interpreting the coefficients, especially \(a\), students gain insights into rates of change in the vertical motion, directly relating to acceleration caused by forces like gravity. Understanding these relationships is crucial to solving motion-related problems.

Linear regression

Linear regression is a statistical method used for modeling the relationship between variables. It's especially useful in this exercise for analyzing how horizontal distance depends on time.
By plotting the time \(t\) against horizontal distance \(x\), students can identify a straight-line pattern. Using linear regression helps determine the best-fit line for these data points, represented as \(x = vt + x_0\). This means finding the line that most closely matches the data points by minimizing the differences between the data points and the line itself.
The slope \(v\) of this line, derived through linear regression, indicates the initial speed or the constant velocity over time in the horizontal direction. Conducting linear regression provides precise mathematical backing for such observations.

Gravity

Gravity plays an essential role in the vertical motion of projectiles. It's the constant force pulling objects towards the Earth's center, giving them acceleration in the downward direction.
In the context of projectile motion, gravity influences the vertical path of an object like the plastic ball in our exercise. This makes vertical motion nonlinear, as opposed to the linearity of horizontal motion without any extraneous forces.
Through quadratic regression analysis of the vertical distance data, students can extract information about gravity. Specifically, the coefficient \(a\) in the vertical motion's quadratic equation is associated with gravitational acceleration. Doubling this coefficient gives the acceleration due to gravity \(g\), illustrating gravity's unbiased effect on the falling ball.