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Unformatted text preview: Physics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocity and Acceleration in 3-D We have defned the velocity and acceleration oF a particle as the frst and second time derivatives oF the position, in the special case oF one dimensional motion. What is the generalization to motion in two or three dimensions? Simply that the position must now be described by a vector r ( t ) and the velocity an acceleration are derivatives with respect to time oF this vector: Velocity and Acceleration Velocity: v = d r dt Velocity and Acceleration Acceleration: a = d v dt = d 2 r dt 2 Here the derivative has its usual defnition. or example: d r dt = lim t r ( t + t ) r ( t ) t . As always, a vector equation is three equations, one for each component . Thus v = d r dt means v x = dx / dt v y = dy / dt v z = dz / dt Constant Velocity or Constant Acceleration In one dimension the Formula For position oF a particle moving at constant velocity was Found earlier to be x = x + vt , where x is the position at t = . PHY 53 1 Kinematics 2 Generalizing this to two or three dimensions is easy: replace the position and velocity by the corresponding vectors: Constant velocity (general) r = r + v t This vector equation is, as always, three component equations: x = x + v x t y = y + v y t z = z + v z t The two vectors r and v lie in a particular plane. If we choose that to be the x-y plane, then the z components of all the vectors in the vector equation will be zero, and we can ignore the third component equation altogether. This is an example of how choosing the coordinate system can often simplify the calculations. Applying the same procedure to the formulas for constant acceleration, we obtain: Constant Acceleration (General) r ( t ) = r + v t + 1 2 a t 2 v ( t ) = v + a t The third formula we found in one dimension, relating speed directly to position, involves the scalar product of two vectors, to be discussed later. Gravity Near Earth's Surface As everyone knows, an object dropped from rest at a point slightly above the earths surface falls down; i.e., it experiences an acceleration directed toward the surface. Galileo discovered by experiment two remarkable facts about this: 1. If the effects of the air and other possible inFuences are neglected, the acceleration is constant. 2. The acceleration is the same for all objects....
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