* In terms of x and y, a x and a y can be expressed as x (m) Note that in one dimension, the velocity and the acceleration of an object are always along the same straight line (either in the same direction or in the opposite direction). However, for motion in two or three dimensions, velocity and acceleration vectors may have any angle between ° and ° between them. Example . The position of a particle is given by .
t ˆ . . where t is in seconds and the coefficients have the proper units for r to be in metres. (a) Find v ( t ) and a ( t ) of the particle.
(b) Find the magnitude and direction of v ( t ) at t = . s. Answer ( ) t . .
ɵ a = . m s – along y - direction At t = . s, . .
v = i + It’s magnitude is - . m s and direction is - = tan ° ≅ with x -axis. . MOTION IN A PLANE WITH CONSTANT ACCELERATION Suppose that an object is moving in x-y plane and its acceleration a is constant.
Over an interval of time, the average acceleration will equal this constant value. Now, let the velocity of the object be v at time t = and v at time t . Then, by definition ( .33a) In terms of components : ox oy ( .33b) Let us now find how the position r changes with time. We follow the method used in the one- dimensional case.
Let r o and r be the position vectors of the particle at time and t and let the velocities at these instants be v o and v . Then, over this time interval t , the average velocity is ( v o + v )/ . The displacement is the average velocity multiplied by the time interval : Fig. .
The average acceleration for three time intervals (a) ∆ t , (b) ∆ t , and (c) ∆ t , ( ∆ t > ∆ t > ∆ t ). (d) In the limit ∆ t g , the average acceleration becomes the acceleration. . .