Find the magnitudes of velocities vA and vB in Figure 3.52 (FIGURE CAN'T COPY)

Name: Problem 10, Chapter 3: College Physics
Uploaded: 2020-06-22T20:44:11-08:00
Duration: 8 min 14 s
Channel: Jordan Vanevery
Description: Find the magnitudes of velocities vA and vB in Figure 3.52 (FIGURE CAN'T COPY)

Find the magnitudes of velocities vA and vB in Figure 3.52...

Key Concepts

Relative Velocity

This concept involves determining the velocity of one object with respect to another by examining the difference or sum of their velocity vectors. It is a core idea in mechanics, allowing analysis of the motion of various bodies from different frames of reference and understanding how these motions relate to one another.

Vector Decomposition

Breaking down a vector into its perpendicular components is essential in many mechanics problems. By resolving velocity into its horizontal and vertical components, one can simplify the analysis and later recombine these components to determine the total magnitude and direction of the vector.

Magnitude Calculation

The overall speed or magnitude of a velocity vector, derived from its components, is usually found using the Pythagorean theorem. This process involves taking the square root of the sum of the squares of its individual perpendicular components, ensuring that the resulting value represents the true magnitude of the velocity.

Transcript

00:02 In this example, given this diagram here with this total velocity vector labeled v -tote that's in red, we want to try to determine what are the magnitudes of the individual vectors that make it up.

00:20 So we know that v total here is just the vector sum of va plus vb, where this is actually va here.

00:37 So i think the easiest way to do this problem, instead of breaking things down into components and adding, is just going to be using the law of signs.

00:56 So the law of signs relates sides and angles to each other of any triangle.

01:05 Specifically, it states something like this.

01:08 If we had a triangle like this, in fact, we'll just use our triangle here, where this is v.

01:18 Total this is vb and this is v a and we'll call this angle uh total we'll make this angle b and make this angle a actually let's just make that whole thing bigger maybe like that so v total theta total vb theta b, and then va on this side, and theta a on this side.

02:10 Okay.

02:11 So the law of signs would state that the ratio of the length of a side to the sign of the angle opposite from it is equal to that of any other.

02:30 So in our triangle, the magnitude of v total over the sign of theta total would be equal to the magnitude of va over the sign of theta suba, which would also be equal to the magnitude of vb over the sign of theta b.

02:57 So if that's the case, we know what v total is going to be, and we can easily determine what theta total is going to be because we have theta a and theta b.

03:10 And if that's the case, as long as we have one of our angles, which we'll have all of them, we'll be able to figure out with the magnitudes of va and vbr.

03:18 So first we'll need to figure out what theta sub t is.

03:25 And the way we're going to do this is that since this is the inside of a triangle, we know that all of these angles should sum up to be 180 degrees.

03:38 So 180 should be equal to theta a plus theta b plus theta t.

03:44 Or if we solve that for theta t, we'll have 180 minus theta a minus theta b, where our theta a is going to be this angle up here, 23 degrees, and our theta b would be 26 .5 degrees.

04:16 So if we plug those in, we'll get that that wide angle there, theta total, should be 130 .5 degrees.

04:26 And now that we know that and the magnitude of v total, we can just use the law of signs here to calculate the magnitude of any of the other vs here.

04:39 So for va, let's start there.

04:45 So we have v total over the sine of theta total equals va divided by the sine of theta a...

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