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Average acceleration is a measure of how much the velocity of an object changes over a period of time. It is expressed as a vector, which means it has both magnitude and direction. The formula to find average acceleration is given by:

• \( \mathbf{a}_{avg} = \frac{\Delta \mathbf{v}}{\Delta t} \)

where \( \Delta \mathbf{v} \) is the change in velocity and \( \Delta t \) is the time interval.
In the given exercise, the dog experiences an average acceleration with a magnitude of 0.45 \( \text{m/s}^2 \) and a specific direction of 31 degrees. To break this down into comprehensible pieces, it is important to understand that acceleration can be dissected into its components using trigonometry.
The horizontal (\(a_x\)) and vertical (\(a_y\)) components of acceleration can be found by:

• \( a_x = a \cos(31^\circ) \)
• \( a_y = a \sin(31^\circ) \)

These allow us to calculate how much each individual component of velocity is affected over the time interval.

Understanding velocity components is essential for solving problems that involve motion in two dimensions, such as the movement of the dog in the open field. Velocity is also a vector, which means it has both magnitude and direction. It can be separated into two perpendicular components – one along the x-axis and one along the y-axis.
Initially, the dog's velocity components are given as \(v_{x}=2.6 \ \text{m/s}\) and \(v_{y}=-1.8 \ \text{m/s}\). When the dog experiences acceleration, these components change over time. The change in the velocity components from \(t_1\) to \(t_2\) can be calculated using the acceleration components and the duration of time over which they act:

• \( \Delta v_x = a_x \Delta t \)
• \( \Delta v_y = a_y \Delta t \)

Here, \( \Delta t = 10 \ \text{s} \). Knowing the initial velocities and the changes, we can find the new velocities at \(t_2\):

• \( v_{x,t2} = v_{x,t1} + \Delta v_x \)
• \( v_{y,t2} = v_{y,t1} + \Delta v_y \)

In this example, the velocity in the y-direction increases, illustrating how the dog’s upward acceleration affects its motion.

Vector analysis is a powerful tool in physics that allows us to mathematically describe how objects in motion move and change direction. Vectors not only have magnitude, but they also have a direction, which is crucial for fully understanding two-dimensional motion.
To analyze vectors, such as velocity vectors, you can visualize them as arrows in a coordinate system. The length of the arrow represents the magnitude, while the direction of the arrow indicates the vector's direction. In this problem, the velocity vectors at \(t_1\) and \(t_2\) highlight how the dog’s movement evolves over time.
Vectors can be added and subtracted not by simple arithmetic, but by looking at their components along each axis. For the dog’s velocity at \(t_2\), once we have both component values \(v_{x,t2}\) and \(v_{y,t2}\), we can compute the overall speed, or resultant velocity, using the Pythagorean theorem:

• \( v_{t2} = \sqrt{v_{x,t2}^2 + v_{y,t2}^2} \)

Moreover, the direction of this new vector is found using:

• \( \theta_{t2} = \tan^{-1}\left(\frac{v_{y,t2}}{v_{x,t2}}\right) \)

This approach shows how motion can be broken down and comprehensively understood using vector analysis.