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uppose that you are in a school bus that is traveling at a velocity of 8 m/s in a positive direction You walk with a velocity of 3 m/s toward the front of the bus If a friend of yours is standing on the side of the road watching the bus with you on it go by, how fast would your friend say that you are moving If the bus is traveling at 8 m/s, this means that the velocity of the bus is 8 m/s, as measured by your friend in a coordinate system fixed to the road When you are standing still, your velocity relative to the road is also 8 m/s, but your velocity relative to the bus is zero Walking at 3 m/s toward the front of the bus means that your velocity is measured relative to the bus The problem can be rephrased as follows: Given the velocity of the bus relative to the road and your velocity relative to the bus, what is your velocity relative to the road A vector representation of this problem is shown in Figure 6-9a After studying it, you will find that your velocity relative to the street is 11 m/s, the sum of 8 m/s and 3 m/s Suppose that you now walk at the same speed toward the rear of the bus What would be your velocity relative to the road Figure 6-9b shows that because the two velocities are in opposite directions, the resultant velocity is 5 m/s, the difference between 8 m/s and 3 m/s You can see that when the velocities are along the same line, simple addition or subtraction can be used to determine the relative velocity Take a closer look at how these results were obtained and see if you can find a mathematical rule to describe how velocities are combined in these relative-velocity situations For the above situation, you can designate the velocity of the bus relative to the road as vb/r, your velocity relative to the bus as vy/b, and the velocity of you relative to the road as vy/r To find the velocity of you relative to the road in both cases, you vectorially added the velocities of you relative to the bus and the bus relative to the road Mathematically, this is represented as vy/b vb/r vy/r The more general form of this equation is as follows Relative Velocity va/b vb/c va/c.

Figure 6-9 When a coordinate system is moving, two velocities are added if both motions are in the same direction and one is subtracted from the other if the motions are in opposite directions

The relative velocity of object a to object c is the vector sum of object a s velocity relative to object b and object b s velocity relative to object c

test=> select current_date, current_time, current_timestamp; date | timetz | now + + 2006-08-15 | 19:51:31837-04 | 2006-08-15 19:51:31837-04 (1 row) test=>

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Phillipe whirls a stone of mass m on a rope in a perfect horizontal circle above his head such that the stone is at a height, h, above the ground The circle has a radius of r, and the tension in the rope is T Suddenly the rope breaks and the stone falls to the ground The stone travels a horizontal distance, s, from the time the rope breaks until it impacts the ground Find a mathematical expression for s in terms of T, r, m, and h Does your expression change if Phillipe is walking 050 m/s relative to the ground

This method for adding relative velocities also applies to motion in two dimensions For example, airline pilots cannot expect to reach their destinations by simply aiming their planes along a compass direction They must take into account the plane s speed relative to the air, which is given by their airspeed indicators, and their direction of flight relative to the air They also must consider the velocity of the wind at the altitude they are flying relative to the ground These two vectors must be combined, as shown in Figure 6-10, to obtain the velocity of the airplane relative to the ground The resultant vector tells the pilot how fast and in what direction the plane must travel relative to the ground to reach its destination A similar situation occurs for boats traveling on water with a flowing current

Relative Velocity of a Marble Ana and Sandra are riding on a ferry boat that is traveling east at a speed of 40 m/s Sandra rolls a marble with a velocity of 075 m/s north, straight across the deck of the boat to Ana What is the velocity of the marble relative to the water

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