Question

A $13-\mathrm{g}$ goldfinch has a speed of $8.5\mathrm{m}/\mathrm{s}$. What is its kinetic energy?

Short answer

The kinetic energy of the goldfinch is approximately 0.47 J.

Step-by-step solution

Introduction to Kinetic Energy Formula

Kinetic energy is given by the formula: $KE=\frac{1}{2}m{v}^2$, where $KE$ is the kinetic energy, $m$ is the mass in kilograms, and $v$ is the velocity in meters per second.

Convert Mass to Kilograms

The goldfinch's mass is given in grams, so we need to convert it to kilograms. Since there are 1000 grams in a kilogram, we have: $m=\frac{13}{1000}=0.013\text{ kg}$.

Calculating the Kinetic Energy

Now, substitute the values for the mass $m=0.013\text{ kg}$ and velocity $v=8.5\text{ m/s}$ into the kinetic energy formula: $KE=\frac{1}{2}\times 0.013\times (8.5{)}^2$.

Carrying out the Calculation

First, calculate $(8.5{)}^2=72.25$. Then find the product: $0.013\times 72.25=0.93925$. Finally, multiply by $\frac{1}{2}$: $KE=\frac{1}{2}\times 0.93925=0.469625$.

Final Result

The kinetic energy of the goldfinch is approximately $0.469625\text{ J}$. Rounding to appropriate precision, the kinetic energy is $0.47\text{ J}$.

Key concepts

Mass Conversion

When working with physics problems, especially those involving kinetic energy, it's often necessary to ensure all units are compatible. Mass conversion is a key step in ensuring this compatibility. Often, measurements may be given in grams, but the standard unit for mass in the International System of Units (SI) is kilograms. For example, if you have a mass of 13 grams, you will need to convert it to kilograms.

• To convert from grams to kilograms, simply divide the mass by 1000.
• In this case, 13 grams becomes 0.013 kilograms after conversion.

By converting mass correctly, you can avoid errors in calculations and ensure your physics formulas work correctly.

Velocity

Velocity is a vector quantity that refers to both speed and direction of an object's movement. In the case of calculating kinetic energy, we focus on the speed aspect, which is how fast the object is moving regardless of direction. It’s measured in meters per second (m/s). Knowing an object's velocity is crucial as it is directly squared in the kinetic energy formula.

• For instance, our example involves a bird moving at 8.5 m/s.
• Remember, even slight increases in velocity can significantly increase kinetic energy, since velocity is squared in the calculation.

Having accurate velocity measurements ensures precision in energy calculations.

Physics Formula

Physics relies on formulas to express complex concepts in understandable ways. One such essential physics formula is the kinetic energy formula: $$KE=\frac{1}{2}m{v}^2$$

• Here, $KE$ represents the kinetic energy.
• The term $m$ stands for mass in kilograms, and $v$ is velocity in meters per second.

This formula reveals how kinetic energy is affected by both mass and velocity. Remember, because velocity is squared, it plays a larger role in increasing kinetic energy. Always ensure to substitute the variables with correct units for accurate results.

Energy Calculation

To calculate an object's kinetic energy, you need to substitute known values into the kinetic energy formula, which is $KE=\frac{1}{2}m{v}^2$. Here's how to approach it:

• First, ensure that the mass is expressed in kilograms and the velocity in meters per second.
• Then, substitute these values into the formula. For example, with a mass of 0.013 kg and a velocity of 8.5 m/s, calculate as follows:
  1. Square the velocity: ${8.5}^2=72.25$.
  2. Multiply by the mass: $0.013\times 72.25=0.93925$.
  3. Multiply the result by $\frac{1}{2}$: $0.469625$.
• The final result gives the kinetic energy in joules, which in this case is approximately 0.47 J after rounding.

Careful calculation ensures understanding and accuracy, emphasizing the relationship between an object's mass, velocity, and its kinetic energy.