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When dealing with velocities in two dimensions, it's important to know the overall speed, or the magnitude of the velocity. This is where the Pythagorean theorem comes in handy. Imagine you have two velocity components: one along the x-axis and another along the y-axis. To find the overall velocity, treat these components as the sides of a right triangle.

According to the Pythagorean theorem, the length of the hypotenuse (resultant velocity) is the square root of the sum of the squares of these components. In our example with initial velocities, you'd calculate this as:

• Initial x-component = 6.00 m/s
• Initial y-component = -2.00 m/s
• Magnitude of velocity = \(\sqrt{(6.00)^2 + (-2.00)^2}\)

So, the magnitude of the velocity gives you a single value representing the object's speed in both directions together.

Velocity, simply put, is an object's speed and direction. In physics, we often break this down into components, especially when working with two or three dimensions. These components are vectors pointing in specific directions. For example, a velocity vector in 2D can be split into two components: one along the x-axis and one along the y-axis.

In our given exercise, the object starts with a velocity that has an x-component of 6.00 m/s and a y-component of -2.00 m/s. The signs indicate direction; positive x means rightwards, and negative y means downwards. When the velocity changes, its components become 8.50 m/s and 5.00 m/s. This indicates a new direction and speed; positive values here mean movement to the right and upwards.

By looking at the components separately, you can easily understand how the object's motion changes in each direction.

Kinetic energy is the energy that an object possesses due to its motion. It is calculated using the formula \(K = \frac{1}{2} m v^2\), where \(m\) is the mass of the object, and \(v\) is its velocity. In our problem, kinetic energy is calculated before and after a change in velocity.

First, you find the initial kinetic energy using the magnitude of the initial velocity (from the Pythagorean theorem) and the mass, then you do the same for the new, or final, velocity.

Let's break it down:

• Initial KE: \(\frac{1}{2} \times 5.75 \times (\text{initial velocity magnitude})^2\)
• Final KE: \(\frac{1}{2} \times 5.75 \times (\text{final velocity magnitude})^2\)

Finally, the change in kinetic energy is simply the difference between the final and initial kinetic energies. This value tells you how much the energy has increased or decreased due to the change in velocity.