Rank Velocity Vectors by Kinetic Energy

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To rank the given velocities by kinetic energy, first calculate the magnitude of each velocity vector, as kinetic energy is proportional to the square of this magnitude. The formula K = 1/2mv^2 indicates that only the speed (magnitude) matters, not the direction. The magnitudes for the velocities are determined as follows: v = 5i has the highest magnitude, followed by v = 4i + 3j, v = -3i + 4j, v = -4i + 3j, and v = 3i - 4j, with v = 5 m/s at 30 degrees being calculated using trigonometric components. Understanding that kinetic energy depends solely on speed allows for an accurate ranking based on these magnitudes. The discussion emphasizes the importance of recognizing the distinction between vector direction and the scalar quantity of speed when calculating kinetic energy.
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Rank the following velocities according to the kinetic energy a particle will have with each velocity, greatest first: (a) v = 4i +3j, (b) v = -4i +3j, (c) v = -3i + 4j, (d) v = 3i - 4j, (e) v = 5i, and (f) v = 5 m/s at 30 degrees to the horizontal.



K = 1/2mv^2



I am not sure how to start.

I am sure this question is rather easy considering it is the very first question after the chapter. However, I read the chapter 2x and they never mention finding kinetic energy given velocity vectors... Maybe someone could explain to me how I should start this problem or what the key ideas here are. When I see velocity vectors I see points on a graph, maybe I should be looking at them differently?
 
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Velocity is a vector, so it has a direction and it has a magnitude. The latter is commonly known as speed. Kinetic energy is proportional to the square of magnitude.
 

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