Introduction
In physics, understanding how different properties of motion are interconnected is crucial. Two of the most important concepts are Kinetic Energy (K) and Momentum (p). While they describe different aspects of a moving object, they are mathematically related through the object's mass.
The Mathematical Relationship
To solve the problem, we must first establish the relationship between kinetic energy and momentum.
- The standard formula for Kinetic Energy is: $K = \frac{1}{2}mv^2$
- The formula for Momentum is: $p = mv$
If we manipulate these equations to eliminate velocity ($v = \frac{p}{m}$), we substitute it back into the kinetic energy equation:
$$K = \frac{1}{2}m(\frac{p}{m})^2$$ $$K = \frac{1}{2}m(\frac{p^2}{m^2})$$ $$K = \frac{p^2}{2m}$$
Step-by-Step Solution
Question: How does the kinetic energy of a body change if its momentum is halved?
Given:
- Let initial momentum be $p_1 = p$
- Let initial kinetic energy be $K_1 = \frac{p^2}{2m}$
- Let final momentum be $p_2 = \frac{p}{2}$ (halved)
Calculation: Substitute $p_2$ into the energy equation to find the new kinetic energy $K_2$:
$$K_2 = \frac{(p_2)^2}{2m}$$ $$K_2 = \frac{(\frac{p}{2})^2}{2m}$$ $$K_2 = \frac{\frac{p^2}{4}}{2m}$$ $$K_2 = \frac{1}{4} \times \frac{p^2}{2m}$$
Since $K_1 = \frac{p^2}{2m}$, we can write:
$$K_2 = \frac{1}{4}K_1$$
Conclusion: The new kinetic energy will be one-fourth (1/4) of the original kinetic energy.
Intuition
Because kinetic energy depends on the square of the momentum ($K \propto p^2$), any change in momentum is magnified. If you divide the momentum by a factor of 2, the energy is divided by $2^2 = 4$. This is why high-momentum objects carry significantly more destructive potential than low-momentum objects.