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Kinetic energy reflects the energy that an object possesses due to its motion. It is an essential concept in physics that helps us understand dynamics and motion. The formula to calculate kinetic energy is given as \[ KE = \frac{1}{2}mv^2 \]where:

• \( KE \) represents the kinetic energy in joules (\( \,J \,\)).
• \( m \) is the mass of the object in kilograms (\( \, kg \,\)).
• \( v \) is the velocity of the object in meters per second (\( \, m/s \,\)).

This formula forms the basis for calculating the kinetic energy possessed by an object when both its mass and velocity are known. By understanding this equation, you can solve various problems related to kinetic energy. In our example, knowing that the boulder's kinetic energy is 305 J and velocity is 12.4 m/s, we can move forward with calculations to determine its mass, and predict changes when velocity adjusts.

To determine the mass of an object when given its kinetic energy and velocity, you can rearrange the kinetic energy formula to solve for mass. The equation becomes:\[ m = \frac{2KE}{v^2}\]This lets you find mass by isolating it on one side of the equation. For our specific exercise:

• \( KE = 305 \, J \)
• \( v = 12.4 \, m/s \)

Substituting these values into the equation, we calculate \[m = \frac{2 \times 305}{12.4^2}\]Solving, we find that the mass is approximately 3.98 kg. This illustrates how mass can be derived if you know the energy of motion and the velocity of the object. Understanding how to rearrange the formula is crucial for solving real-world and textbook physics problems efficiently.

Velocity plays a crucial role in determining kinetic energy. When velocity changes, kinetic energy changes exponentially because it is proportional to the square of velocity in the kinetic energy formula: \[ KE = \frac{1}{2}mv^2 \] If you double the velocity, the kinetic energy increases by a factor of four. Here’s why:

• New velocity \( v' = 2v \)
• New kinetic energy becomes \( KE' = \frac{1}{2}m(2v)^2 = 4 \times \frac{1}{2}mv^2 = 4KE \)

In the given boulder exercise, when the speed doubled to 24.8 m/s, the kinetic energy increased to 1220 J, four times the original. Alternatively, if you halve the velocity, kinetic energy decreases to a quarter:

• New velocity \( v'' = \frac{v}{2} \)
• New kinetic energy \( KE'' = \frac{1}{2}m(\frac{v}{2})^2 = \frac{1}{4} \times \frac{1}{2}mv^2 = \frac{1}{4}KE \)

Thus, halving the speed to 6.2 m/s resulted in reducing kinetic energy to 76.25 J, illustrating the squared relationship between velocity and kinetic energy.