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S E C T I O N 4 . 2 • Two-Dimensional Motion with Constant Acceleration 81 where x, y, and r change with time as the particle moves while the unit vectors iˆ and jˆ remain constant. If the position vector is known, the velocity of the particle can be ob- (4.7) Because a is assumed constant, its components a x and a y also are constants. Therefore, we can apply the equations of kinematics to the x and y components of the velocity vec- xf " v xi & a x t and v yf " v yi & a y t into Equation 4.7 to determine the final velocity at any time t, we obtain (4.8) This result states that the velocity of a particle at some time t equals the vector sum of v i and the additional velocity at acquired at time t as a result of con- stant acceleration. It is the vector version of Equation 2.9. Similarly, from Equation 2.12 we know that the x and y coordinates of a particle moving with constant acceleration are Substituting these expressions into Equation 4.6 (and labeling the final position r f ) gives (4.9) which is the vector version of Equation 2.12. This equation tells us that the position r f is the vector sum of the original position r i , a displacement v i t arising from the initial velocity of the particle and a displacement at 2 resulting from the constant acceleration of the particle. Graphical representations of Equations 4.8 and 4.9 are shown in Figure 4.5. Note from Figure 4.5a that v f is generally not along the direction of either v i or a because the relationship between these quantities is a vector expression. For the same reason, 1 2 r f " r i & v i t & 1 2 at 2 " (x i iˆ & y i jˆ) & (v xi iˆ & v yi jˆ)t & 1 2 (a x iˆ & a y jˆ)t 2 r f " (x i & v xi t & 1 2 a x t 2 )iˆ & (y i & v yi t & 1 2 a y t 2 )jˆ x f " x i & v xi t & 1 2 a x t 2
y f " y i & v yi t & 1 2 a y t 2 v f " v i & at " (v xi iˆ & v yi jˆ)& (a x iˆ & a y jˆ)t v f " (v xi & a x t) iˆ & (v yi & a y t) jˆ v " dr dt " dx dt iˆ & dy dt
jˆ " v x iˆ & v y jˆ y x a y t v yf v yi v f v i at v xi a x t v xf (a) y x y f y i r f v i t v xi t x f (b) a y t 2 1 2 v yi t r i at 2 1 2 a x t 2 1 2 x i Active Figure 4.5 Vector representations and components of (a) the velocity and (b) the posi- tion of a particle moving with a constant acceleration a. Velocity vector as a function of time Position vector as a function of time At the Active Figures link at http://www.pse6.com, you can investigate the effect of different initial positions and velocities on the final position and velocity (for constant acceleration). |