Three Chapter 3

• Motion is always described with respect to a reference point. To illustrate this, consider a boat on the ocean in the South Pacific. From the deck of the boat an island can be seen in the distance with a palm tree on it. It is noticed that after a time, the direction of sight to the island has changed, and the island appears to be further away. Is the island moving?
  • To determine if the island is moving, we need a reference frame. If we choose the reference frame to be the boat, then the island is moving. If we choose the reference frame to be the island, then the boat is moving and not the island. If we choose the reference frame to be the water, then possibly both island and boat are moving.
  • Another more familiar example is where you are on a bus which is stopped at a red light. Another bus is beside the bus you are on. As you look at the other bus, you get the impression that your bus is moving backwards. Upon looking out the window in the opposite direction, you realize that the other bus was moving forward instead of you moving backwards. This experience happens because you first chose the other bus as your reference frame, and you noticed you were moving relative to it. When you looked out the other window, you switched reference frames to the road, and now you were not moving but the other bus was moving with respect to the road.
• Motion - If a line joining a point to the reference frame changes in length and/or direction, then motion occurs.
• Reference Frame (Point) - A place from which measurements are made.
• If an object moves with respect to a moving reference frame, then this further confuses the issue. If you are in an automobile moving at 100 km/h [N], and you throw a ball out of the window at 50 km/h [N] with respect to the car, what is the velocity of the ball with respect to the ground?
  • To answer this question, one simply adds the two velocities together to get the ball's velocity with respect to the ground (150 km/h [N]).
  • One must realize that the answer asked for here is with respect to a stationary reference frame. The velocity of the car is given with respect to this stationary reference frame.
  • The car's velocity is basically the velocity of a moving reference frame since the ball's velocity is stated with respect to the car.
  • To get the answer of 150 km/h [N], we simply used the following formula.

• V_s in the formula stands for the velocity with respect to a stationary reference frame, v_m stands for the velocity of the moving reference frame (In the example, the car's velocity with respect to the ground.), and v is the velocity of the object with respect to the moving reference frame (In the example, the ball's velocity with respect to the car).
• In general, the formula applies to the following reference frame set up.

• It must be noted that in the example, all velocities were in the same direction. If they are not, then vector addition must occur using components and trigonometry.
• An airplane can flying at 300 km/h with respect to the air. The pilot aims her plane northward to fly to the desired location. A wind blows at 80 km/h from [W 30^o S]. What is the ground speed of the airplane?
  • To answer this question, one must define the variables with respect to the above equation. In this example, V_s is what is being asked for. Since the plane's velocity is given wrt (with respect to) the air, and the air is moving, then the velocity of the air must be the velocity of the moving frame (i.e. v_m ). That leaves the velocity of the plane being v. The given velocities are not along the same line, therefore vector addition is needed using trigonometry. The solution goes as follows.

• This problem can also be done by using Sine and Cosine Law as follows.

• Whenever only two vectors are to be added, using sine and cosine laws may make your work easier. When more than two vectors are to be added, sine and cosine laws may confuse you and so the first method (the component method) is best.
• You should keep in mind, the component method will work in all situations. Using vector diagrams to solve these kinds of problems is discouraged because it is not very accurate.

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October 29, 2013