Objects A and B are at the origin. If object A moves to $(- 2, 2)$ $(-2 ,2 )$ and object B moves to $(7, - 5)$ $(7 ,-5 )$ over $3 s$ $3 s$ , what is the relative velocity of object B from the perspective of object A? Assume that all units are denominated in meters.

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Objects A and B are at the origin. If object A moves to $(- 2, 2)$ $(-2 ,2 )$ and object B moves to $(7, - 5)$ $(7 ,-5 )$ over $3 s$ $3 s$ , what is the relative velocity of object B from the perspective of object A? Assume that all units are denominated in meters.

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Answer 1 · 2017-05-25T15:42:23

$3.80 \frac{m}{s}$ $3.80 "m"/"s"$ , ${37.9}^{o}$ $37.9^o$ south of the horizontal

Explanation:

The components of the velocity of object A are

$v_{A x} = \frac{- 2 m}{3 s} = - 0.667 \frac{m}{s}$ $v_(Ax) = (-2"m")/(3 "s") = -0.667 "m"/"s"$

$v_{A y} = \frac{2 m}{3 s} = 0.667 \frac{m}{s}$ $v_(Ay) = (2"m")/(3 "s") = 0.667 "m"/"s"$

and object B

$v_{B x} = \frac{7 m}{3 s} = 2.33 \frac{m}{s}$ $v_(Bx) = (7"m")/(3 "s") = 2.33 "m"/"s"$

$v_{B y} = \frac{- 5 m}{3 s} = - 1.67 \frac{m}{s}$ $v_(By) = (-5 "m")/(3"s") = -1.67 "m"/"s"$

We're trying to find the velocity of object B with respect to object A. We'll call this velocity $v_{B / A}$ $v_(B"/"A)$ , the velocity of object B with respect to the origin $v_{B / O}$ $v_(B"/"O)$ , and the velocity of object A relative to the origin $v_{A / O}$ $v_(A"/"O)$ .

The equation for relative velocity, using these frames of reference, is

${\to v}_{A / O} = {\to v}_{A / B} + {\to v}_{B / O}$ $vec v_(A"/"O) = vec v_(A"/"B) + vec v_(B"/"O)$

And remembering that ${\to v}_{A / B} = - {\to v}_{B / A}$ $vec v_(A"/"B) = -vec v_(B"/"A)$ , this equation becomes

${\to v}_{A / O} = {\to v}_{B / O} - {\to v}_{B / A}$ $vec v_(A"/"O) = vec v_(B"/"O) - vec v_(B"/"A)$

Since we're solving for ${\to v}_{B / A}$ $vec v_(B"/"A)$ ,

${\to v}_{B / A} = {\to v}_{B / O} - {\to v}_{A / O}$ $vec v_(B"/"A) = vec v_(B"/"O) - vec v_(A"/"O)$

Or, in terms of components,

$v_{B x / A x} = v_{B x / O} - v_{A x / O}$ $v_(Bx"/"Ax) = v_(Bx"/"O) - v_(Ax"/"O)$

$v_{B y / A y} = v_{B y / O} - v_{A y / O}$ $v_(By"/"Ay) = v_(By"/"O) - v_(Ay"/"O)$

Now, let's plug in our known velocity components:

$v_{B x / A x} = 2.33 \frac{m}{s} - - 0.667 \frac{m}{s} = 3.00 \frac{m}{s}$ $v_(Bx"/"Ax) = 2.33"m"/"s" - -0.667"m"/"s" = 3.00 "m"/"s"$

$v_{B y / A y} = - 1.67 \frac{m}{s} - 0.667 \frac{m}{s} = - 2.33 \frac{m}{s}$ $v_(By"/"Ay) = -1.67"m"/"s" - 0.667 "m"/"s" = -2.33 "m"/"s"$

Thus, the magnitude of the velocity of object B with respect to object A is

$v_{B / A} = \sqrt{{(3.00 \frac{m}{s})}^{2} + {(- 2.33 \frac{m}{s})}^{2}} = 3.80 \frac{m}{s}$ $v_(B"/"A) = sqrt((3.00"m"/"s")^2 + (-2.33"m"/"s")^2) = color(red)(3.80 "m"/"s"$

And the direction of the velocity of object B with respect to object A is

$ϕ = arctan (\frac{- 2.33 \frac{m}{s}}{3.00 \frac{m}{s}}) = - {37.9}^{o}$ $phi = arctan ((-2.33"m"/"s")/(3.00 "m"/"s")) = color(blue)(-37.9^o$

There are technically two directions that satisfy this, each opposite to each other in a coordinate plane (the other angle is thus $142^{o}$ $142^o$ ). We have to choose the one that points from object A to object B, which if you were to plot their coordinates, would indeed be $- {37.9}^{o}$ $-37.9^o$ .

The direction is more easily found in this case, as we also could have just found the inverse tangent of the differences in the $y$ $y$ -positions over the difference in the $x$ $x$ -position of the two objects:

$ϕ = arctan (\frac{2 m - - 5 m}{- 2 m - 7 m}) = - {37.9}^{o}$ $phi = arctan ((2"m"--5"m")/(-2"m"-7"m")) = -37.9^o$

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Objects A and B are at the origin. If object A moves to $(- 2, 2)$ $(-2 ,2 )$ and object B moves to $(7, - 5)$ $(7 ,-5 )$ over $3 s$ $3 s$ , what is the relative velocity of object B from the perspective of object A? Assume that all units are denominated in meters.

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Explanation:

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Objects A and B are at the origin. If object A moves to (−2,2)(-2 ,2 ) and object B moves to (7,−5)(7 ,-5 ) over 3s3 s, what is the relative velocity of object B from the perspective of object A? Assume that all units are denominated in meters.

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Objects A and B are at the origin. If object A moves to $(- 2, 2)$ $(-2 ,2 )$ and object B moves to $(7, - 5)$ $(7 ,-5 )$ over $3 s$ $3 s$ , what is the relative velocity of object B from the perspective of object A? Assume that all units are denominated in meters.