91Ó°ÊÓ

When we talk about kinetic energy in physics, we use a specific formula to express it. This formula is written as:

• \( KE = \frac{1}{2} m v^2 \)

Here, \( KE \) stands for kinetic energy, \( m \) represents the mass of the object, and \( v \) is the velocity at which the object is moving.
This formula shows that kinetic energy depends on both the mass of the object and its speed. Importantly, because velocity is squared in the formula, changes in speed have a dramatic effect on the kinetic energy.
Doubling the velocity will quadruple the kinetic energy, showing the non-linear relationship nature of speed to kinetic energy. Understanding this formula is the key to solving kinetic energy calculations and discerning the factors that influence an object's energy in motion.

The relationship between mass, velocity, and kinetic energy is fascinating. Instead of acting independently, mass and velocity together determine how much kinetic energy an object will have.

• If the mass is increased while keeping velocity constant, the kinetic energy increases proportionally:
\( KE \propto m \).

Similarly,

• if the velocity increases and the mass remains constant, kinetic energy increases but at a much faster rate:
\( KE \propto v^2 \).

In the given exercise, object B was heavier than object A by a factor of \( 3.77 \) and moved more slowly by a factor of \( 2.34 \).
This meant that its increased mass and decreased velocity had combined effects on its kinetic energy. More mass contributed positively, whereas the slower velocity contributed negatively. However, because velocity is squared in the formula, it can have a larger impact than changes in mass. Hence, it is always essential to balance these two factors when calculating energy.

Calculating kinetic energy involves plugging the known values into the kinetic energy formula and doing some straightforward arithmetic. For object B,

• The kinetic energy calculation utilized the known kinetic energy of object A (13.4 J) and factored in the adjustments in both mass and velocity for object B.

To find object B's kinetic energy:

• First, substitute object B's new mass \( m_B = 3.77m \); and adjusted velocity \( v_B = \frac{v}{2.34} \) into the formula:
\[ KE_B = \frac{1}{2} (3.77m) \left( \frac{v}{2.34} \right)^2 \]
• Using Object A's known kinetic energy \( KE_A = 13.4 \, J \), substitute into the equation:\[ KE_B = 3.77 \times 13.4 \times \left( \frac{1}{2.34} \right)^2 \]
• Calculate the values to finally determine that Object B's kinetic energy is approximately \( 9.204 \, J \).

These steps show how mass and velocity interact mathematically to influence kinetic energy.