Question

Which has the greater kinetic energy, an object with a mass of 2.0 \(\mathrm{kg}\) and a velocity of 1.0 \(\mathrm{m} / \mathrm{s}\) or an object with a mass of 1.0 \(\mathrm{kg}\) and a velocity of 2.0 \(\mathrm{m} / \mathrm{s}\) ?

Short answer

The object with a mass of 1.0 kg and a velocity of 2.0 m/s has greater kinetic energy (2.0 J) than the object with a mass of 2.0 kg and a velocity of 1.0 m/s (1.0 J).

Step-by-step solution

Object 1: Calculate Kinetic Energy

Using the given formula, we can compute the kinetic energy of object 1: \( KE_1 = \frac{1}{2} \times 2.0 \, kg \times (1.0 \, m/s)^2 \)

Object 1: Simplify and Calculate

Now, we simplify the expression and calculate the kinetic energy: \( KE_1 = 1.0\, kg \times 1.0 \, m^2/s^2 = 1.0 \, J \) where J represents joules, the unit of energy.

Object 2: Calculate Kinetic Energy

Similarly, we can compute the kinetic energy of object 2: \( KE_2 = \frac{1}{2} \times 1.0\, kg \times (2.0\, m/s)^2 \)

Object 2: Simplify and Calculate

After simplifying this expression and calculating the kinetic energy: \( KE_2 = 0.5 \, kg \times 4.0 \, m^2/s^2 = 2.0 \, J \)

Compare Kinetic Energies

Now we compare the kinetic energies of both objects: \( KE_1 = 1.0 \, J \) \( KE_2 = 2.0 \, J \) Since \( KE_2 > KE_1 \), the object with a mass of 1.0 kg and a velocity of 2.0 m/s has greater kinetic energy than the object with a mass of 2.0 kg and a velocity of 1.0 m/s.

Key concepts

Understanding Mass in Kinetic Energy

Mass is one of the key components when we talk about kinetic energy. It refers to the amount of matter in an object, which directly affects how much kinetic energy it can have. When you have an object, the mass is measured in kilograms (kg).
Kinetic energy is calculated using the formula:

• \( KE = \frac{1}{2}mv^2 \)

Here, \( m \) stands for mass. It acts as a multiplier of energy, meaning the heavier the object, the more energy it can carry.
To understand it better, consider an object with more mass using the same speed. It will possess more kinetic energy because the mass is larger, even if the velocity stays the same.
Imagine a bowling ball versus a tennis ball rolling at the same pace. The bowling ball has more mass, thus more kinetic energy.

The Role of Velocity in Kinetic Energy

Velocity is another crucial factor in the kinetic energy equation. It's not just speed; it refers to how fast and in what direction an object moves. Velocity is measured in meters per second (m/s).
In the kinetic energy formula:

• \( KE = \frac{1}{2}mv^2 \)

The \( v \) stands for velocity and it has a squared relationship to kinetic energy. This means a small change in velocity can have a big impact on energy. If you double the velocity, the kinetic energy increases by four times because you square the velocity in the formula.
If you think about riding a bike, when you pedal faster (increase your velocity), you feel much more energy flowing. In the context of our objects, the second object has a higher velocity, leading to more kinetic energy, even with a lighter mass.

Energy Units Explained: Understanding Joules

Energy is measured in specific units to help quantify it, and in physics, kinetic energy is often expressed in joules. A joule (J) is a derived unit that denotes the amount of energy transferred when a force of one newton moves an object one meter.
In the context of kinetic energy, both the mass and velocity are converted into the joule through the formula \( KE = \frac{1}{2}mv^2 \). A higher joule count signifies more energy.

• For instance, if an object has a kinetic energy of 2.0 J, it means it has more energy than an object with 1.0 J.
  

With our two objects, the object with the higher energy in joules is the one with greater kinetic energy. It's as if you're filling the same-size bottle with more water; more joules mean more kinetic energy.