Become four times
Explanation
Kinetic energy ($K.E.$) is the energy possessed by an object by virtue of its motion. It is mathematically expressed as:
$K.E. = \frac{1}{2}mv^2$
where $m$ is the mass of the object and $v$ is its velocity. This formula indicates that kinetic energy is directly proportional to the mass and directly proportional to the square of the velocity.
Analysis
To determine the change in kinetic energy when velocity is doubled, we compare the initial and final states:
- Initial State: Let the initial velocity be $v$. The initial kinetic energy is $E_1 = \frac{1}{2}mv^2$.
- Final State: The velocity is doubled, so the new velocity is $v' = 2v$.
- Calculation: Substituting the new velocity into the formula:
$E_2 = \frac{1}{2}m(2v)^2$
$E_2 = \frac{1}{2}m(4v^2)$
$E_2 = 4 \times \left(\frac{1}{2}mv^2\right)$
$E_2 = 4 E_1$
Therefore, the new kinetic energy is four times the original kinetic energy.
Key Takeaway:
Kinetic energy depends on the square of the velocity ($K.E. \propto v^2$). Consequently, doubling the speed of an object quadruples its energy, which has significant implications for impact force and braking distances in mechanics.