the average acceleration as the time interval ∆ t goes to zero : lim a lim The acceleration of an object at a particular time is the slope of the velocity-time graph at that instant of time. For uniform motion, acceleration is zero and the x-t graph is a straight line inclined to the time axis and the v-t graph is a straight line parallel to the time axis. For motion with uniform acceleration, x-t graph is a parabola while the v-t graph is a straight line inclined to the time axis. .
The area under the velocity-time curve between times t and t is equal to the displacement of the object during that interval of time. . For objects in uniformly accelerated rectilinear motion, the five quantities, displacement x , time taken t , initial velocity v , final velocity v and acceleration a are related by a set of simple equations called kinematic equations of motion : v = v + at 2ax if the position of the object at time t = is . If the particle starts at x = x , x in above equations is replaced by ( x – x ).
POINTS TO PONDER . The origin and the positive direction of an axis are a matter of choice. You should first specify this choice before you assign signs to quantities like displacement, velocity and acceleration. .
If a particle is speeding up, acceleration is in the direction of velocity; if its speed is decreasing, acceleration is in the direction opposite to that of the velocity. This statement is independent of the choice of the origin and the axis. . The sign of acceleration does not tell us whether the particle’s speed is increasing or decreasing.
The sign of acceleration (as mentioned in point ) depends on the choice of the positive direction of the axis. For example, if the vertically upward direction is chosen to be the positive direction of the axis, the acceleration due to gravity is negative. If a particle is falling