You're heading an international effort to save Earth from an asteroid heading toward us at \(15 \mathrm{km} / \mathrm{s}\). Your team mounts a rocket on the asteroid and fires it for 10 min, after which the asteroid is moving at \(19 \mathrm{km} / \mathrm{s}\) at \(28^{\circ}\) to its original path. In a news conference, what do you report for the acceleration imparted to the asteroid?

Vector subtraction is a fundamental concept in physics when dealing with quantities that have both magnitude and direction. In the case of our asteroid deflection scenario, the change in velocity caused by the rocket is described using vector subtraction.

To perform vector subtraction, we consider each vector's components along the coordinate axes. A vector in two dimensions has an x-component and a y-component. In our problem, the initial velocity of the asteroid is given as a vector pointing along the negative x-axis, and after the rocket's intervention, the velocity changes. To find the change, we calculate the final velocity components using trigonometry: the cosine of the angle gives us the x-component and the sine gives us the y-component.

Subtracting the initial velocity's components from the final velocity's components provides us with the change in velocity vector represented by \( \Delta v_x \) and \( \Delta v_y \). These components describe how the direction and speed of the asteroid have been altered due to the application of force over a period.

Acceleration is the rate at which an object changes its velocity. To find the acceleration, we start by determining the change in velocity, which we have already outlined in the previous section by performing vector subtraction.

The change in velocity over time gives us the acceleration. In our example, time is provided as 10 minutes. Since we need to use consistent units for accurate calculation, we convert this duration to seconds, resulting in 600 seconds. We then divide each component of the change in velocity by this time to get the respective components of the acceleration vector: \( a_x = \frac{\Delta v_x}{t} \) and \( a_y = \frac{\Delta v_y}{t} \).

With the x and y components of acceleration, we can determine the overall magnitude of the acceleration by using the Pythagorean theorem. By taking the square root of the sum of the squares of \( a_x \) and \( a_y \) components, we find the asteroid's acceleration magnitude due to the rocket's force.

Proper conversion of units is crucial in physics to ensure that equations and calculations are consistent and meaningful. Whenever we're calculating quantities like velocity, acceleration, or force, we must pay attention that all our measurements are in the appropriate units.

In the exercise, we have velocities in kilometers per second (\(\mathrm{km/s}\)) and time in minutes. To calculate acceleration, which is typically expressed in meters per second squared (\(\mathrm{m/s^2}\)) or kilometers per second squared (\(\mathrm{km/s^2}\)), we need time in seconds. By converting 10 minutes to 600 seconds, we maintain unit consistency in our calculations.

It's essential to remember that incorrect unit conversions can lead to errors in the result. Therefore, vigilance in unit conversion is not just a pedantic concern, but a necessary part of ensuring accurate and reliable physics calculations. As a best practice, always check your units before, during, and after performing your calculations to guarantee they align with the physical quantities you're working with.