Understanding Kinetic Energy Change

This question asks us to find the percentage increase in the kinetic energy of an object when its velocity is tripled. We need to use the formula for kinetic energy and calculate the initial and final values to determine the change.

Formula for Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion. The formula for kinetic energy is given by:

$$ KE = \frac{1}{2}mv^2 $$

Where:

• $m$ is the mass of the object.
• $v$ is the velocity of the object.

Step-by-Step Calculation

Initial Conditions:

Mass ($m$) = 10 kg

Initial Velocity ($v_1$) = 5 m/s

Calculating Initial Kinetic Energy ($KE_1$):

Using the formula $KE = \frac{1}{2}mv^2$:

$$ KE_1 = \frac{1}{2} \times (10 \text{ kg}) \times (5 \text{ m/s})^2 $$

$$ KE_1 = \frac{1}{2} \times 10 \times 25 $$

$$ KE_1 = 5 \times 25 $$

$$ KE_1 = 125 \text{ Joules} $$

Final Conditions:

The velocity of the object is tripled. So, the new velocity ($v_2$) is:

$$ v_2 = 3 \times v_1 $$

$$ v_2 = 3 \times 5 \text{ m/s} $$

$$ v_2 = 15 \text{ m/s} $$

The mass remains the same: $m = 10$ kg.

Calculating Final Kinetic Energy ($KE_2$):

Using the formula $KE = \frac{1}{2}mv^2$ with the new velocity:

$$ KE_2 = \frac{1}{2} \times (10 \text{ kg}) \times (15 \text{ m/s})^2 $$

$$ KE_2 = \frac{1}{2} \times 10 \times 225 $$

$$ KE_2 = 5 \times 225 $$

$$ KE_2 = 1125 \text{ Joules} $$

Calculating the Change in Kinetic Energy ($\Delta KE$):

The change in kinetic energy is the difference between the final and initial kinetic energy:

$$ \Delta KE = KE_2 - KE_1 $$

$$ \Delta KE = 1125 \text{ J} - 125 \text{ J} $$

$$ \Delta KE = 1000 \text{ Joules} $$

Calculating the Percentage Change in Kinetic Energy:

The percentage change is calculated as:

$$ \text{Percentage Change} = \frac{\text{Change in KE}}{\text{Initial KE}} \times 100\% $$

$$ \text{Percentage Change} = \frac{\Delta KE}{KE_1} \times 100\% $$

$$ \text{Percentage Change} = \frac{1000 \text{ J}}{125 \text{ J}} \times 100\% $$

$$ \text{Percentage Change} = 8 \times 100\% $$

$$ \text{Percentage Change} = 800\% $$

Conclusion

When the velocity of the object is tripled, the kinetic energy increases significantly because it depends on the square of the velocity. The calculation shows that the change in kinetic energy is an increase of 800%.