If the linear momentum of a moving object gets doubled due to application of a force, then its kinetic energy will
to increase by four times
Understanding Kinetic Energy and Momentum
This question asks about the relationship between the kinetic energy and linear momentum of a moving object when its momentum is doubled. To solve this, we need to recall the definitions of these two physical quantities and the relationship between them.
Linear Momentum: Linear momentum (\(p\)) is a measure of the mass in motion. It is defined as the product of an object's mass (\(m\)) and its velocity (\(v\)).
\(p = mv\)
Kinetic Energy: Kinetic energy (\(K\)) is the energy possessed by an object due to its motion. It is defined as half of the product of an object's mass (\(m\)) and the square of its velocity (\(v\)).
\(K = \frac{1}{2}mv^2\)
Relating Kinetic Energy and Linear Momentum
We can derive a relationship between kinetic energy and linear momentum. Starting from the kinetic energy formula:
\(K = \frac{1}{2}mv^2\)
We know that \(p = mv\). We can write \(v = \frac{p}{m}\). Substituting this into the kinetic energy formula:
\(K = \frac{1}{2}m\left(\frac{p}{m}\right)^2\)
\(K = \frac{1}{2}m\left(\frac{p^2}{m^2}\right)\)
\(K = \frac{p^2}{2m}\)
This formula, \(K = \frac{p^2}{2m}\), shows that kinetic energy is directly proportional to the square of the linear momentum, assuming the mass of the object remains constant.
Effect of Doubling Linear Momentum on Kinetic Energy
Let the initial linear momentum of the object be \(p_1\) and the initial kinetic energy be \(K_1\). According to the derived relationship:
\(K_1 = \frac{p_1^2}{2m}\)
The question states that the linear momentum gets doubled due to the application of a force. Let the new momentum be \(p_2\). So,
\(p_2 = 2p_1\)
Now, let's find the new kinetic energy, \(K_2\), using the new momentum \(p_2\). Assuming the mass \(m\) of the object remains constant:
\(K_2 = \frac{p_2^2}{2m}\)
Substitute \(p_2 = 2p_1\) into this equation:
\(K_2 = \frac{(2p_1)^2}{2m}\)
\(K_2 = \frac{4p_1^2}{2m}\)
We can rewrite this as:
\(K_2 = 4 \times \left(\frac{p_1^2}{2m}\right)\)
Notice that the term \(\frac{p_1^2}{2m}\) is the initial kinetic energy \(K_1\). So, we can write:
\(K_2 = 4K_1\)
This means the new kinetic energy is four times the initial kinetic energy. Therefore, the kinetic energy increases by four times when the linear momentum is doubled.
Conclusion on Kinetic Energy Increase
When the linear momentum of a moving object is doubled while its mass remains constant, its kinetic energy increases by a factor of four. This is because kinetic energy is proportional to the square of the momentum.
| Quantity | Initial State | Final State (Momentum Doubled) | Change |
|---|---|---|---|
| Linear Momentum (p) | \(p_1\) | \(p_2 = 2p_1\) | Doubled |
| Kinetic Energy (K) | \(K_1 = \frac{p_1^2}{2m}\) | \(K_2 = \frac{p_2^2}{2m} = \frac{(2p_1)^2}{2m} = 4 \frac{p_1^2}{2m} = 4K_1\) | Increased by four times |
Revision Table: Linear Momentum and Kinetic Energy
| Concept | Definition/Formula | Relationship |
|---|---|---|
| Linear Momentum | \(p = mv\) | \(K = \frac{p^2}{2m}\) |
| Kinetic Energy | \(K = \frac{1}{2}mv^2\) | \(p = \sqrt{2mK}\) |
Additional Information on Momentum and Energy
Linear momentum is a vector quantity (it has direction), while kinetic energy is a scalar quantity (it only has magnitude). In an isolated system, the total linear momentum is conserved (Law of Conservation of Momentum). However, kinetic energy may or may not be conserved in a process, especially in collisions (e.g., inelastic collisions).
The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. A force applied over a distance causes a change in kinetic energy. The application of a force also causes a change in momentum over time (Impulse-Momentum theorem).
This problem highlights how proportional relationships, particularly those involving squares, can significantly impact physical quantities. Doubling a quantity that is squared in a formula results in the final value being four times the original.
Important Questions from Momentum and Energy
- What would be the momentum of the bullet and the gun before firing?
Which among the following has the same dimension as that of energy?
Light with an energy flux of 500 kW/m2 falls for 5 minutes at normal incidence on a non-reflecting circular surface with a radius of 10 cm. The total momentum delivered to this surface has a magnitude of ______.
