Figure 2: Time-dependent vectors
For acceleration, replace r by v in the above expressions, v dv = : t dt
a = lim
t!0
(25)
In Cartesian coordinates
6
b a = vxbi + vybj + vz k; dvx b dvy b dvz b dv a = = i+ j+ k; dt dt dt dt d2 xb d2 y b d2 z b = i + 2 j + 2 k: dt2 dt dt
(26) (27) (28)
Motion in one dimension
Consider a particle of mass, m, moving along the positive x-axis as in …g 3.
Figure 3: Motion along x-axis The velocity is positive for motion in sense of x increasing and negative for x decreasing. In time dt distance travelled by particle is dx = vdt. In the …nite time interval between t1 , when position of particle is x1 , and time t2 when position is x2 , distance travelled is Z t2 v dt: (29) s = (x2 x1 ) = t1
This is represented by the shaded area in …g 4(a). We need to know how v varies with t in order to calculate s.
4