Relative Velocities

RELATIVE VELOCITIES Relative-velocity problems are problems involving the relationship between the velocity of an object as seen by two different observers (or frames of reference). For example, a passenger on a moving training looking out the window seeing a person tossing a ball. The velocity of the ball as seen by the rider in the train will be related to the velocity of the ball as seen by the person tossing the ball. The two velocities are related through the relative velocity of the two persons (or frames of reference). Let the person on the train be observer A, the person tossing the ball be observer B. and let VBA be the velocity of B as seen by A, then the velocity of the ball as seen by A, i.e. VOA, is related to the velocity of the ball as seen by B, i.e. VOB, by the relative velocity equation Note the structure of the notation: (a) the first subscript of the velocity denotes the object and the second subscript denotes the observer (b) the right side of the equation has the intermediate observer B in between the object O and the observer A. The relative velocity equation is a vector addition equation. Given any two velocities the third velocity can be obtained. Reciprocity equation: If the observer and object are switched then the velocity changes sign, i.e. That is the velocity of A as seen by B is the opposite of the velocity of B as seen by A. Example 1: A boat’s velocity in still water is 5.0m/s at 35 o north of East. The ocean current is 1.2m/s at 28o west of North. What is the velocity of the boat relative to the shore? First we must identify the two observers (or frames of reference): The first frame of reference is the ocean or water (W), and the second frame of reference is the shore or Earth (E). Next we must identify the object: The object is the boat (B). Next we must identify the three velocities: The first velocity is the boat’s velocity in still water. This is the boat’s velocity as seen by the water or VBW. The second velocity is the ocean current which is the velocity of the water as seen by the Earth or VWE. The third velocity is the velocity of the boat as seen by the shore or Earth or VBE. These velocities are related by the relative velocity equation Now it is time to get numbers for the velocities. 1 The standard coordinate system is East is +x and North is +y. So the polar coordinates of VBW is VBW = (5.0m/s, 35o), and the polar coordinate of VWE is VWE = (1.2m/s, 118o), see figure below. Note that 28o west of North means the angle is 28o to the left of the y axis (or North), but the polar angle is measured relative to the x axis (or East). Therefore we must add 90o to 28o to get 118o. N or y VBW VWE 28o 35o E or x The last step is to calculate the third velocity VBE from the vector equation: The x component equation: The y component equation: Convert to polar coordinates The velocity of the boat relative to the shore is VBE = (5.3m/s, 48o), i.e. 5.3m/s at 48o north of East. 2 Example 2: Car A has a velocity of 18m/s at 62o north of West, and car B has a velocity of 22m/s at 49o south of East. What is the velocity of car B relative to car A? The velocity of each car is relative to the ground or Earth (E), i.e. VAE and VBE. What is asked for is the velocity of B relative to A, i.e. VBA. The relative velocity equation is Notice that this equation requires VEA, but we are given VAE. Here we have to use the reciprocity relation So the relative velocity equation becomes The velocities of the cars are VAE = (18m/s, 180o – 62o = 118o) and VBE = (22m/s, 360o – 49o = 311o), see figure below. N or y VAE 62o E or x 49o VBE The x component equation: The y component equation: 3 Convert to polar coordinates The velocity of car B relative to car A is VBA = (40m/s, 125o), i.e. 40m/s at 55o north of West. 4 Exercise 1: A plane’s velocity relative to the air (called its airspeed) is 376km/h at 29 o north of East. If there is a wind blowing North at 95km/h, what is the plane’s velocity relative to the ground (called its ground speed)? The object is the plane (P). The two frames of reference are the air/wind (A) and the ground (G). Which of the following velocities is the airspeed of the plane: VPG, VPA, VGP or VAP? Circle the correct answer. Which of the following velocities is the ground speed of the plane: VPG, VPA, VGP or VAP? Which of the following velocities is the wind speed: VPG, VPA, VGP or VAP, VAG, or VGA? Draw the vectors representing the airspeed of the plane and the wind speed below. N or y E or x The polar coordinates of the airspeed is (fill in the boxes with appropriate letters or numbers) Solve the relative equation for the components of the unknown velocity 5 Exercise 2: The pilot of plane A sees a plane B approaching his plane at a speed of 255km/h at 67o south of West. If the ground speed of plane A is 312km/h at 45o north of East, what is the ground speed of plane B? The object is plane B (B). The two frames of reference are plane A (A) and the ground (G). Which of the following velocities is the ground speed of plane A: VAG, VGA, VAB or VBA? Circle the correct answer. 6 Which of the following velocities is the ground speed of plane B: VBG, VGB, VAB or VBA? Which of the following velocities is the velocity of plane B relative to plane A: VAB, VBA? Draw the vectors representing the airspeed of the plane and the wind speed below. N or y E or x The polar coordinates of the ground speed of plane A is (fill in the boxes with appropriate letters or numbers) Solve the relative equation for the components of the unknown velocity 7 8