- Define the kinetic energy formula \(K = \frac{1}{2}mv^2\) and assume a constant velocity \(v = 2\) m/s, simplifying the relationship to \(K = 2m\).
- Calculate the kinetic energy for the specified mass values: \(5, 25, 50, 75, 100\) kg, resulting in \(10, 50, 100, 150, 200\) Joules.
- Present these results in the requested table format.
- Recognize that the relationship \(K(m) = 2m\) creates a linear graph, confirming that kinetic energy scales proportionally with mass at a constant velocity, yielding the final values: \(\boxed{10, 50, 100, 150, 200}\).
Explanation
- Problem Analysis and Relationship Definition
The relationship between mass (\(m\)) and kinetic energy (\(K\)) of an object moving at a constant velocity (\(v\)) is governed by the kinetic energy formula:
\[K = \frac{1}{2}mv^2\]
To complete the requested table, we need a constant velocity. Let us assume a velocity of \(v = 2\) m/s. This allows us to calculate the kinetic energy for any given mass using the simplified expression:
\[K = \frac{1}{2} \cdot m \cdot (2)^2 = \frac{1}{2} \cdot m \cdot 4 = 2m\]
This shows that for a constant velocity of \(2\) m/s, the kinetic energy in Joules is exactly twice the numerical value of the mass in kilograms.
- Calculation and Table Completion
Using the formula \(K = 2m\), we can now calculate the kinetic energy for each of the given mass values:
- For \(m = 5\) kg: \(K = 2 \times 5 = 10\) J
- For \(m = 25\) kg: \(K = 2 \times 25 = 50\) J
- For \(m = 50\) kg: \(K = 2 \times 50 = 100\) J
- For \(m = 75\) kg: \(K = 2 \times 75 = 150\) J
- For \(m = 100\) kg: \(K = 2 \times 100 = 200\) J
The completed table is as follows:
| Mass (kg) | Kinetic Energy (J) |
| :--- | :--- |
| 5 | 10 |
| 25 | 50 |
| 50 | 100 |
| 75 | 150 |
| 100 | 200 |
- Graphing the Relationship
Since the kinetic energy \(K\) is directly proportional to the mass \(m\) (represented by the linear equation \(K = 2m\)), the graph of kinetic energy versus mass is a straight line passing through the origin \((0,0)\). As mass increases, the kinetic energy increases at a constant rate, specifically with a slope of \(2\) J/kg. This linear relationship is characteristic of systems where velocity remains unchanged.
- Final Answer
We have determined the relationship between mass and kinetic energy using a constant velocity of \(2\) m/s. The resulting values are summarized in the table, and the relationship is a linear function \(K(m) = 2m\). The final kinetic energy values for the given masses are \(\boxed{10, 50, 100, 150, 200}\) Joules.
Examples
Understanding the relationship between mass and kinetic energy is vital in fields like automotive safety engineering. For instance, if a car of a certain mass is traveling at a constant speed, engineers must calculate the energy involved in a potential collision to design safety features like crumple zones and airbags. By realizing that kinetic energy scales linearly with mass (\(K \propto m\)), they can predict how heavier vehicles (like trucks) require more robust safety systems than lighter vehicles (like compact cars) even when traveling at the same speed.