Question

Calculate the kinetic energy of an object with a mass of \(1.0 \times\) \(10^{-5} \mathrm{~g}\) and a velocity of \(2.0 \times 10^{5} \mathrm{~cm} / \mathrm{s}\)

Short answer

The kinetic energy of the object with a mass of \(1.0 \times 10^{-8} \mathrm{~kg}\) and a velocity of \(2.0 \times 10^{3} \mathrm{~m/s}\) is \(8.0 \times 10^{-2} \mathrm{~J}\) (Joules).

Step-by-step solution

Convert the given mass and velocity to standard units

To convert mass from grams to kilograms, we will divide by 1000. To convert velocity from centimeters per second to meters per second, we will divide by 100. Mass in kg: \(1.0 \times 10^{-5}\mathrm{~g} = 1.0 \times 10^{-5}\mathrm{~g} \times \frac{1\mathrm{~kg}}{1000\mathrm{~g}} = 1.0 \times 10^{-8}\mathrm{~kg}\) Velocity in m/s: \(2.0 \times 10^{5}\mathrm{~cm/s} = 2.0 \times 10^{5}\mathrm{~cm/s} \times \frac{1\mathrm{~m}}{100\mathrm{~cm}} = 2.0 \times 10^{3}\mathrm{~m/s}\)

Calculation of kinetic energy

Now that we have the mass and velocity in standard units, we can use the kinetic energy formula: K.E. = 0.5 * m * v^2 Substitute the values: K.E. = 0.5 * \(1.0 \times 10^{-8}\mathrm{~kg}\) * \((2.0 \times 10^{3}\mathrm{~m/s})^2\)

Simplify and compute the kinetic energy

Calculate the square of the velocity and then multiply by the mass and 0.5. K.E. = 0.5 * \(1.0 \times 10^{-8}\mathrm{~kg}\) * \((4.0 \times 10^{6}\mathrm{~(m/s)^2})\) K.E. = \(2.0 \times 10^{-8}\mathrm{~kg} \times 4.0 \times 10^{6}\mathrm{~(m/s)^2}\) K.E. = \(8.0 \times 10^{-2}\mathrm{~J}\) (Joules) The kinetic energy of the object is 8.0 x 10^{-2} Joules.

Key concepts

Unit Conversion

When working with physical quantities, it's often necessary to convert units to standard forms. This step is crucial as it ensures consistency and accuracy.

In our problem, we start with the mass given in grams and the velocity in centimeters per second. To convert these:

• Mass: Convert from grams to kilograms. Since 1 kg equals 1000 g, divide the mass by 1000. Our mass of 1.0 x 10^-5 g becomes 1.0 x 10^-8 kg.
• Velocity: Change from cm/s to m/s. Knowing that 1 m is 100 cm, divide the velocity by 100. So, 2.0 x 10^5 cm/s converts to 2.0 x 10^3 m/s.

This standardization allows us to apply formulas and comprehend results in commonly accepted scientific units, like Joules for kinetic energy.

Kinetic Energy Formula

Kinetic energy is the energy an object possesses because of its motion. Calculating it involves the kinetic energy formula: \(KE = \frac{1}{2} m v^2\). Here:

• \(m\) represents mass in kilograms (kg).
• \(v\) is velocity in meters per second (m/s).

Using this, we find the object's kinetic energy by substituting the converted mass and velocity into the formula. This calculation captures how both mass and speed contribute to motion energy. In essence, even small mass gains significant energy at high velocities.

Physics

Physics, the science of matter and energy, relies heavily on formulas like those for kinetic energy to describe and predict natural phenomena. Understanding how objects move and interact involves:

• Principles of motion, like Newton's laws, impacting how energy manifests.
• Units such as Joules, which measure energy levels consistently.
• Conservative systems, meaning energy doesn't vanish but changes form.

In calculating kinetic energy, physics principles help turn data into applicable insights about energy transformations and the forces affecting motion.

Mass and Velocity

Mass and velocity are fundamental quantities in physics, directly influencing kinetic energy. Let's delve into each:

• Mass: The amount of matter in an object, measured in kilograms. Greater mass means more energy required for movement, thus yielding more kinetic energy for the same velocity.
• Velocity: This vector quantity combines speed and direction. As velocity squared appears in the kinetic energy formula, an increase in velocity has a much larger effect on energy than an equivalent increase in mass.

In practice, these concepts explain why lighter but faster objects sometimes have comparable or superior energy levels to heavier, slower ones. Understanding these factors reveals how different scenarios affect movement and energy priorities.