Question

Consider a rabbit that is at \(x=8.1 \mathrm{~m}\) at \(t=0\) and moves with a constant velocity of \(-1.6 \mathrm{~m} / \mathrm{s}\). What is the equation of motion for the rabbit?

Short answer

The equation of motion is \(x(t) = 8.1 - 1.6t\).

Step-by-step solution

Understand the Problem

We have a rabbit starting at a position of 8.1 meters at time \(t=0\) and moving with a constant velocity of \(-1.6\,\text{m/s}\). We need to find an equation that describes the rabbit's position at any time \(t\).

Identify the Appropriate Equation

For an object moving with constant velocity, the equation of motion is given by \(x(t) = x_0 + vt\), where \(x_0\) is the initial position and \(v\) is the velocity.

Substitute the Known Values

We know that the initial position \(x_0 = 8.1\,\text{m}\) and the velocity \(v = -1.6\,\text{m/s}\). Substitute these values into the equation: \(x(t) = 8.1 + (-1.6)\cdot t\).

Simplify the Equation

The equation becomes \(x(t) = 8.1 - 1.6t\). This is the equation of motion for the rabbit.

Key concepts

constant velocity

In physics, understanding constant velocity is crucial for solving many motion-related problems. When an object moves with constant velocity, it means that its speed and direction remain unchanged throughout its motion.
This implies that the object covers equal distances in equal amounts of time.
Key characteristics of constant velocity include:

• The speed or magnitude remains the same.
• The direction of motion does not change.

In our example, the rabbit is moving with a constant velocity of \( -1.6 \, \mathrm{m/s} \).
This indicates the rabbit is moving in a straight line backward at a steady pace.
Understanding constant velocity is fundamental in forming equations of motion, which describe how the position of an object changes over time.

initial position

The initial position, often denoted as \( x_0 \), is the starting point of an object in motion. It is essential for determining the object's future positions using the equation of motion.
The initial position provides a reference point from which all subsequent motion is measured.
For the rabbit in the exercise, the initial position is given as \( x_0 = 8.1 \, \mathrm{m} \) at time \( t = 0 \).
This means the rabbit begins its journey 8.1 meters from a chosen reference point, typically from the origin.
Knowing the initial position allows us to understand where the object is starting from, which is crucial in calculating its position at any future time using known velocities and times.

position-time relation

The position-time relation describes how the position of an object changes over time. It is typically expressed as an equation that connects position \( x(t) \), time \( t \), initial position \( x_0 \), and velocity \( v \).
In the context of an object moving with constant velocity, the equation of motion is: \[x(t) = x_0 + vt\]This equation tells us that the position of the object at any time \( t \) is dependent on its starting point and the product of velocity and time.
For the rabbit moving backward with velocity \( -1.6 \, \mathrm{m/s} \), the equation becomes:\[x(t) = 8.1 - 1.6t\]This simple linear equation lets us predict where the rabbit will be at any given time just by substituting different values of \( t \).
It showcases a practical example of how position and time are intricately related through constant velocity.

physics problem solving

Solving a physics problem involves a systematic approach to understanding and applying concepts to find a solution. When tackling motion problems like the one with our rabbit, it is important to:

• Identify the given information and what needs to be found.
• Choose the correct physics principles and formulas applicable to the problem.
• Substitute known values into the equation and simplify where needed.

For our rabbit problem, we needed to find the equation of motion.
We started by noting the initial position and velocity, then chose the correct motion equation.
By substituting the known values, we derived the simple linear equation: \( x(t) = 8.1 - 1.6t \).
This methodical approach can be applied to any physics problem to make the solution easier and more accurate. It also helps develop a deeper understanding of how different physics concepts connect and interact.