Relative Velocity in Two Dimensions We can now apply these concepts to describing motion in two dimensions. Consider a particle Pand reference frames Sand S', as shown in the figure. The position of the origin of SY as measured in Sisi's's, the position of Pas measured in S' is f pg' , and the position of Pas measured in Sis Fps. y' y P S Iss. TPS S x z The positions of particle P relative to frames S and S' are Tips and I'ps', respectively. From the figure we see that The relative velocities are the time derivatives of the position vectors. Therefore, VPS = Vps +Vs's. The velocity of a particle relative to S is equal to its velocity relative to SI plus the velocity of S' relative to S. We can extend this to any number of reference frames. For particle P with velocities VPA, VPB, and Vpc in frames A, B, and C. VPC = VPA + VAB + VBC. We can also see how the accelerations are related as observed in two reference frames by differentiating the velocities: åps = aps +äs's· We see that if the velocity of S' relative to Sis a constant, then as's = 0 and äps = äps . This says the acceleration of a particle is the same as measured by two observers moving at a constant velocity relative to each other.
Motion of a Car Relative to a Truck A truck is traveling south at a speed of 70 km/h toward an intersection. A car is traveling east toward the intersection at a speed of 80 km/h (see below). What is the velocity of the car relative to the truck? N W E S 70 km/h VTE 80 km/h VCE A car travels east toward an intersection while a truck travels south toward the same intersection. Strategy First, we must establish the reference frame common to both vehicles, which is Earth. Then, we write the velocities of each with respect to the reference frame of Earth, which enables us to form a vector equation that links the car, the truck, and Earth to solve for the velocity of the car with respect to the truck. Solution The velocity of the car with respect to Earth is VCE = 80 km/h i. The velocity of the truck with respect to Earth is VTE = - 70 km/h j. Using the velocity addition rule, the relative motion equation we are seeking is VCT = VCE + VET·
Here, Vor is the velocity of the car with respect to the truck, and Earth is the connecting reference frame. Since we have the velocity of the truck with respect to Earth, the negative of this vector is the velocity of Earth with respect to the truck: VET = - VTR. The vector diagram of this equation is shown below: N VCT - VCE + VET W E S VCT VET 0 CE Vector diagram of the vector equation VCT = VCR +