Question

Two dragonflies have the following equations of motion: $$\begin{array}{rl}& x_1=2.2\mathrm{m}+(0.75\mathrm{m}/\mathrm{s})t\\ & x_2=-3.1\mathrm{m}+(-1.1\mathrm{m}/\mathrm{s})t\end{array}$$ (a) Which dragonfly is moving faster? (b) Which dragonfly starts out closer to $x=0$ at $t=0$ ?

Short answer

(a) The second dragonfly is moving faster. (b) The first dragonfly starts closer to $x=0$.

Step-by-step solution

Find Speeds of Dragonflies

To determine which dragonfly is moving faster, look at the coefficients of the time variable $t$ in both equations. These coefficients represent the speed of each dragonfly. For $x_1=2.2\text{m}+(0.75\text{m/s})t$, the speed is $0.75\text{m/s}$. For $x_2=-3.1\text{m}+(-1.1\text{m/s})t$, the speed is $1.1\text{m/s}$, albeit in the negative direction. The absolute value of speed, which represents the magnitude, is greater for the second dragonfly.

Compare Constant Terms for Initial Positions

To determine which dragonfly starts closer to $x=0$ at $t=0$, examine the constant terms in the equations of motion. For $x_1=2.2\text{m}+(0.75\text{m/s})t$, at $t=0$, $x_1=2.2\text{m}$. For $x_2=-3.1\text{m}+(-1.1\text{m/s})t$, at $t=0$, $x_2=-3.1\text{m}$. Compare the absolute values of these positions to see which is closer to zero: $2.2\text{m}$ is closer to zero than $-3.1\text{m}$.

Key concepts

Equations of Motion

In physics, the equations of motion are fundamental for analyzing the movement of objects. When looking at these equations, they describe how an object's position changes over time. The general form can be seen as: $$x=x_0+vt$$ where $x$ is the position, $x_0$ is the initial position, and $v$ is the velocity. In the context of our dragonfly problem, we have:

• Dragonfly 1: $x_1=2.2\text{m}+(0.75\text{m/s})t$
• Dragonfly 2: $x_2=-3.1\text{m}+(-1.1\text{m/s})t$

These equations allow us to analyze not only where each dragonfly starts their journey but also their velocity. By examining the coefficients and constant terms, students can identify how swiftly each object is moving and where they began compared to a reference point. This foundational knowledge helps us solve many real-world physics problems involving motion.

Speed Calculation

Calculating speed involves extracting the velocity component from the equation of motion. The speed of an object is the magnitude of its velocity, and it indicates how fast the object is travelling, irrespective of its direction. From our equations:

• Dragonfly 1's velocity: $0.75\text{m/s}$
• Dragonfly 2's velocity: $-1.1\text{m/s}$

Even though Dragonfly 2's velocity is negative, which indicates direction, its speed is found by taking the absolute value, resulting in $1.1\text{m/s}$. This is faster than Dragonfly 1's $0.75\text{m/s}$. Thus, the key takeaway is that when comparing speeds, focus on the absolute values, which give us the speed rather than directional information.

Initial Position Analysis

Initial position analysis is crucial in determining where each object starts its journey. The equations of motion provide these initial positions as the constant terms when $t=0$. For the dragonflies:

• Dragonfly 1's initial position: $2.2\text{m}$
• Dragonfly 2's initial position: $-3.1\text{m}$

These initial positions tell us each dragonfly's starting point relative to $x=0$. By comparing absolute values, $2.2\text{m}$ is closer to zero than $-3.1\text{m}$. This tells us that Dragonfly 1 starts closer to the reference point. Initial position analysis is especially useful in determining how far an object has to travel to change certain conditions, such as reaching a target or a minimum distance, making it an essential skill in physics.