Question

What is the kinetic energy of a particle of mass \(m\) moving through space with velocity \(v ?\)

Short answer

The kinetic energy (KE) of a particle with mass \(m\) and velocity \(v\) can be calculated using the formula \(KE = \frac{1}{2} \times m \times v^2\). Substitute the given values of mass and velocity into the formula and solve to find the kinetic energy.

Step-by-step solution

1) Write down the given information

We are given the mass (m) of the particle and the velocity (v) with which it moves through space.

2) Write the Kinetic Energy formula

The formula for calculating the Kinetic Energy (KE) of a particle is: \(KE = \frac{1}{2} \times m \times v^2\)

3) Substitute the given values into the formula

Now, let's substitute the given values of mass (m) and velocity (v) into the kinetic energy formula: \(KE = \frac{1}{2} \times m \times v^2\)

4) Calculate the Kinetic Energy

Finally, solve for KE by multiplying the mass and the square of the velocity, and then multiplying the result by 1/2: \(KE = \frac{1}{2} \times m \times v^2\) The result will give the Kinetic energy of the particle with mass m moving through space with velocity v.

Key concepts

Kinetic Energy Formula

Kinetic energy is the energy possessed by a body due to its motion. It's an important concept in physics, as it is one of the forms of energy conserved in an isolated system according to the law of conservation of energy. The kinetic energy formula is integral to understanding how different factors such as mass and speed affect the energy of a moving object.

The equation for kinetic energy (KE) in its simplest form is:
\[ KE = \frac{1}{2} \times m \times v^2 \]
Here, 'm' represents the mass of the object, and 'v' represents the object's velocity. The term \( \frac{1}{2} \) is a constant that appears in the formula.
To gain insight into why this formula takes this form, we can look at the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. This principle leads to the conclusion that an object's kinetic energy is proportional to the square of its velocity and directly proportional to its mass.

Mass and Velocity in Physics

Mass and velocity are key components not only in the kinetic energy formula but in many other principles of physics as well. Mass is a fundamental property of an object that doesn't change regardless of location and is mostly an indication of how much 'stuff' is in the object. The greater an object's mass, the more it resists acceleration when a force is applied, a concept known as inertia.

Velocity, on the other hand, is a vector quantity that describes an object's speed and the direction of its motion. It's important to note that velocity can change even if the speed remains constant – a change in direction means a change in velocity. As velocity is squared in the kinetic energy formula, even small increases in velocity can lead to significant increases in an object's kinetic energy, making it a critical variable for calculations involving motion.

Calculating Kinetic Energy

Computing the kinetic energy of an object involves inserting the values of mass and velocity into the kinetic energy equation. As simple as these calculations can seem, they can lead to a deeper understanding of how objects will behave when subjected to different forces and speeds. For example, consider a car and a bicycle traveling at the same velocity; the car, with its greater mass, will have a much higher kinetic energy than the bicycle.

Let's break down the calculation process with an example of an object with a mass of 5 kilograms moving at a velocity of 2 meters per second:

\(KE = \frac{1}{2} \times m \times v^2\)
\(KE = \frac{1}{2} \times 5 \times (2)^2\)
\(KE = 10 \ Joules\)
This calculation shows that the kinetic energy of this 5 kg object moving at 2 m/s is 10 Joules. This exercise illustrates how kinetic energy can be quantified and compared among objects with varying masses and velocities. Understanding these calculations is crucial for fields ranging from engineering to astrophysics, where precise evaluation of motion is required.