# Objects A and B are at the origin. If object A moves to #(-3 ,5 )# and object B moves to #(-6 ,8 )# over #3 s#, what is the relative velocity of object B from the perspective of object A? Assume that all units are denominated in meters.

Finding each object's velocity components is the first thing we can do:

Thus, when expressed similarly, the equation is

These are the opposites of each other:

Thus, the relative velocity equation can be rewritten as

When expressed as component equations, they are

By entering our previously known values, we have

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The relative velocity of object B from the perspective of object A is (-1 m/s, 1 m/s).

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To find the relative velocity of object B from the perspective of object A, we can use the formula for relative velocity, which is the difference in velocities between the two objects.

First, we need to find the velocities of object A and object B. Velocity is given by the change in position divided by the time taken. The positions of object A and object B after 3 seconds are (-3, 5) and (-6, 8) meters, respectively.

The change in position for object A is ( \Delta x_A = -3 , \text{m} ) and ( \Delta y_A = 5 , \text{m} ). Therefore, the velocity of object A is ( v_A = \frac{\Delta x_A}{\Delta t} = \frac{-3 , \text{m}}{3 , \text{s}} = -1 , \text{m/s} ) in the x-direction and ( v_{Ay} = \frac{\Delta y_A}{\Delta t} = \frac{5 , \text{m}}{3 , \text{s}} = \frac{5}{3} , \text{m/s} ) in the y-direction.

Similarly, for object B, the change in position is ( \Delta x_B = -6 , \text{m} ) and ( \Delta y_B = 8 , \text{m} ). Thus, the velocity of object B is ( v_B = \frac{\Delta x_B}{\Delta t} = \frac{-6 , \text{m}}{3 , \text{s}} = -2 , \text{m/s} ) in the x-direction and ( v_{By} = \frac{\Delta y_B}{\Delta t} = \frac{8 , \text{m}}{3 , \text{s}} = \frac{8}{3} , \text{m/s} ) in the y-direction.

The relative velocity of object B from the perspective of object A is then given by the difference in velocities in each direction:

[ v_{\text{rel}} = (v_{Bx} - v_{Ax}, v_{By} - v_{Ay}) = (-2 , \text{m/s} - (-1 , \text{m/s}), \frac{8}{3} , \text{m/s} - \frac{5}{3} , \text{m/s}) ]

Simplifying this gives the relative velocity as ( v_{\text{rel}} = (-1 , \text{m/s}, \frac{3}{3} , \text{m/s}) = (-1 , \text{m/s}, 1 , \text{m/s}) ) in the x and y directions, respectively.

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To find the relative velocity of object B from the perspective of object A, we can use the formula for relative velocity:

[ \text{Relative velocity} = \frac{\text{Displacement of B}}{\text{Time interval}} ]

First, we find the displacement of B from its initial position to its final position:

[ \text{Displacement of B} = \sqrt{(-6 - 0)^2 + (8 - 0)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ meters} ]

Next, we calculate the time interval, which is given as 3 seconds.

Now, we can find the relative velocity:

[ \text{Relative velocity} = \frac{10 \text{ meters}}{3 \text{ seconds}} = \frac{10}{3} \approx 3.33 \text{ meters/second} ]

Therefore, the relative velocity of object B from the perspective of object A is approximately ( 3.33 ) meters per second.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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