Relative Motion & Acceleration: Definitions & Examples

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  • 0:04 Relative Motion
  • 2:37 Relative Accelaration
  • 4:02 Lesson Summary
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Lesson Transcript
Instructor: Matthew Bergstresser
Relative means 'compared to something', whether that be velocity or acceleration. In this lesson, we will investigate relative motion, including the relative velocity of two objects and relative acceleration of a rigid body.

Relative Motion

Have you ever stubbed your toe? That hurts! But, that's nothing compared to a 100-pound weight dropping on your toe. We just compared one thing to another. This is an analogy for what relative motion is, comparing the motion of two objects or points on an object relative to each other.

Imagine driving down the road and being next to another car in another lane at the same velocity. Relative to each other, your velocities are zero. Relative motion is the mathematical comparison of two or more objects' motions. Let's work an example involving velocities.

An aircraft carrier is moving at 4 m/s due east relative to the water. An aircraft moves on the deck at 1 m/s at 30° east of north relative to the water. How fast is the aircraft moving relative to the aircraft carrier?

Since velocity is a vector, we have to resolve all velocity vectors, and draw a sketch of the scenario.


Vector diagram of plane (p) and aircraft carrier (s)
Ex1_sketch


The ship's velocity is in a pure direction (east) so we don't have to resolve it. The aircraft is moving east and north so we have to determine its velocity east and its velocity north. The ''p'' in the subscripts represent plane, and the ''s'' represents ship.


Graphical resolution of aircraft (p) velocity eastward (E) and northward (N)
Ex1_vec_res


We use the trigonometric functions, sine and cosine, to determine the numerical values of its eastward and northward velocities.


The aircraft (p) has two velocities. Blue is the eastward velocity, and green is the northward velocity
Ex_1_vel


As you can see, this results in 0.5 meters per second and approximately 0.87 meters per second.

To determine the velocity of the plane relative to the ship, we have to use a special setup. Before we can write the special setup, we need a reference point common to both objects. In this case, since the water is still, we can use it as our reference point. We'll label it ''w'', and attach this letter to each object's velocities. Our special setup requires this reference point to be sandwiched between the objects. Let's write out the special equation, like the one appearing here:


This equation reads: the velocity of the plane to the ship equals the velocity of the plane to the water plus the velocity of the water to the ship.
vps


So, plugging in our values, we see that the velocity of the plane to the water is:


vpw


So the velocity of the ship to the water is:


vsw


The special equation we set up earlier needs the velocity of the water to the ship, not the velocity of the ship to the water. Since these are inverses of each other, all we have to do is reverse the velocity of the ship to the water value by making it negative! Doing that, we get:


vws


Now we can plug in the values into our special equation.


ex1_final


So, relative to the aircraft carrier, the aircraft is moving -3.5 m/s east, and 0.87 m/s north.

Relative Acceleration

Acceleration is the change in an object's velocity in a specific period of time. Acceleration is related to velocity, but it's an independent variable.

So what about relative acceleration? Relative acceleration is the comparison of the acceleration of two bodies, or for a rigid body, the acceleration of a point on the rigid body in reference to another point on it. A rigid body is an extended object that doesn't flex, such as a plate. Let's look at an example.

A boomerang is thrown into the air. It's spinning as it's translating (as in, flying through the air). What is the acceleration of point A?


boomerang


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