Calculating Object Speed from Kinetic Energy and Mass

This problem asks us to find the speed of an object given its mass and kinetic energy. Kinetic energy is the energy an object possesses due to its motion. The formula relating kinetic energy, mass, and speed is a fundamental concept in physics.

Understanding Kinetic Energy

Kinetic energy ($\text{KE}$) is directly proportional to the mass ($m$) of the object and the square of its velocity or speed ($v$). The standard formula is:

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

In this problem, we are given the kinetic energy and the mass, and we need to find the speed ($v$). We will rearrange the formula to solve for $v$.

Given Information

• Mass ($m$) = 2000 g
• Kinetic Energy ($\text{KE}$) = 100 J

Units Conversion

The standard units for mass in physics calculations like kinetic energy are kilograms (kg), and for kinetic energy, it's Joules (J). Speed is typically in meters per second (m/s). We need to convert the mass from grams to kilograms.

1 kilogram (kg) = 1000 grams (g)

So, the mass in kilograms is:

$m = 2000 \text{ g} \times \frac{1 \text{ kg}}{1000 \text{ g}}$

$m = 2 \text{ kg}$

Rearranging the Kinetic Energy Formula for Speed

We start with the kinetic energy formula:

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

Multiply both sides by 2:

$2 \cdot \text{KE} = mv^2$

Divide both sides by $m$:

$\frac{2 \cdot \text{KE}}{m} = v^2$

Take the square root of both sides to solve for $v$:

$v = \sqrt{\frac{2 \cdot \text{KE}}{m}}$

Calculating the Speed

Now, substitute the given values (with the mass in kg) into the rearranged formula:

$v = \sqrt{\frac{2 \cdot 100 \text{ J}}{2 \text{ kg}}}$

$v = \sqrt{\frac{200 \text{ J}}{2 \text{ kg}}}$

$v = \sqrt{100 \text{ J/kg}}$

Recall that 1 Joule (J) is equal to 1 kg⋅m²/s². So, J/kg is equivalent to (kg⋅m²/s²)/kg = m²/s².

$v = \sqrt{100 \text{ m}^2/\text{s}^2}$

$v = 10 \text{ m/s}$

Summary of Calculation Steps

The calculated speed of the object is 10.0 m/s.

Revision Table: Kinetic Energy and Speed

Additional Information on Energy Concepts

• Kinetic energy is a scalar quantity, meaning it only has magnitude and no direction.
• The unit of energy, the Joule (J), is defined as the work done when a force of one Newton (N) acts over a distance of one meter (m). $1 \text{ J} = 1 \text{ N} \cdot \text{m}$.
• From Newton's second law ($F=ma$), $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$.
• Therefore, $1 \text{ J} = (1 \text{ kg} \cdot \text{m/s}^2) \cdot 1 \text{ m} = 1 \text{ kg} \cdot \text{m}^2/\text{s}^2$. This confirms the unit analysis done during the calculation.
• The relationship between kinetic energy and speed is squared, meaning if you double the speed, the kinetic energy increases by a factor of four.
• Other forms of energy include potential energy (gravitational, elastic), thermal energy, chemical energy, etc.
• The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another in an isolated system.