If the momentum of a body is tripled keeping the mass as constant, its Kinetic Energy will ________.
become nine times its original value
Understanding Kinetic Energy and Momentum Relation
The question asks how the Kinetic Energy of a body changes if its momentum is tripled while keeping the mass constant. To solve this, we need to understand the relationship between Kinetic Energy and Momentum.
Deriving the Relationship between KE and Momentum
Kinetic Energy (\(KE\)) is the energy a body possesses due to its motion. Its formula is given by:
\[ KE = \frac{1}{2}mv^2 \]
where \(m\) is the mass of the body and \(v\) is its velocity.
Momentum (\(p\)) is the product of a body's mass and velocity. Its formula is:
\[ p = mv \]
To find a relationship between \(KE\) and \(p\), we can express velocity \(v\) in terms of \(p\) and \(m\) using the momentum formula:
\[ v = \frac{p}{m} \]
Now, substitute this expression for \(v\) into the Kinetic Energy formula:
\[ KE = \frac{1}{2}m \left(\frac{p}{m}\right)^2 \]
\[ KE = \frac{1}{2}m \left(\frac{p^2}{m^2}\right) \]
\[ KE = \frac{1}{2}\frac{p^2}{m} \]
\[ KE = \frac{p^2}{2m} \]
This derived formula \(KE = \frac{p^2}{2m}\) shows the relationship between Kinetic Energy (\(KE\)) and Momentum (\(p\)). It tells us that Kinetic Energy is directly proportional to the square of the momentum (\(p^2\)) and inversely proportional to the mass (\(m\)).
Analyzing the Change in Kinetic Energy
Let the original momentum of the body be \(p_{original}\) and its original mass be \(m_{original}\). The original Kinetic Energy (\(KE_{original}\)) is therefore:
\[ KE_{original} = \frac{p_{original}^2}{2m_{original}} \]
According to the question, the momentum is tripled, so the new momentum \(p_{new}\) is:
\[ p_{new} = 3 \times p_{original} \]
The mass is kept constant, so the new mass \(m_{new}\) is:
\[ m_{new} = m_{original} \]
Now, let's calculate the new Kinetic Energy (\(KE_{new}\)) using the new momentum and new mass:
\[ KE_{new} = \frac{p_{new}^2}{2m_{new}} \]
Substitute the values \(p_{new} = 3 p_{original}\) and \(m_{new} = m_{original}\) into the equation:
\[ KE_{new} = \frac{(3 p_{original})^2}{2m_{original}} \]
\[ KE_{new} = \frac{9 p_{original}^2}{2m_{original}} \]
We can rewrite this expression by separating the factor 9:
\[ KE_{new} = 9 \times \left(\frac{p_{original}^2}{2m_{original}}\right) \]
Notice that the term inside the parentheses is the formula for \(KE_{original}\). So, we can write:
\[ KE_{new} = 9 \times KE_{original} \]
This means the new Kinetic Energy is nine times the original Kinetic Energy.
Summary of Changes
| Quantity | Original Value | New Value | Change Factor |
|---|---|---|---|
| Momentum (\(p\)) | \(p_{original}\) | \(p_{new} = 3 p_{original}\) | 3 |
| Mass (\(m\)) | \(m_{original}\) | \(m_{new} = m_{original}\) | 1 (constant) |
| Kinetic Energy (\(KE\)) | \(KE_{original}\) | \(KE_{new} = 9 KE_{original}\) | 9 |
When the momentum of a body is tripled while its mass remains constant, its Kinetic Energy becomes nine times its original value.
Revision Table: Kinetic Energy and Momentum
| Concept | Formula | Relationship Summary |
|---|---|---|
| Kinetic Energy (KE) | \(KE = \frac{1}{2}mv^2\) | Proportional to mass and square of velocity. |
| Momentum (p) | \(p = mv\) | Proportional to mass and velocity. |
| KE and Momentum Relation | \(KE = \frac{p^2}{2m}\) | KE is proportional to the square of momentum and inversely proportional to mass. |
Additional Information: Energy and Momentum Concepts
Understanding Kinetic Energy and Momentum is fundamental in physics. Here are some related concepts:
- Conservation of Momentum: In a closed system (where no external forces act), the total momentum remains constant. This principle is crucial in analyzing collisions and explosions.
- Work-Energy Theorem: The work done on an object is equal to the change in its kinetic energy. This theorem connects the concepts of work, force, and energy.
- Types of Energy: Besides kinetic energy, there are other forms like potential energy (gravitational, elastic), thermal energy, chemical energy, etc. The total energy in a closed system is conserved (Conservation of Energy).
- Vector vs. Scalar: Momentum is a vector quantity (having both magnitude and direction), while Kinetic Energy is a scalar quantity (having only magnitude).
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