Displacement is just a change in position. It does not depend on actual path length.
In rectilinear motion, the magnitude of position is expressed as a number and direction is shown by + or – sign.
The magnitude of displacement may be equal to or less than actual path length.
In a course of motion, the magnitude of the displacement may be zero, but the corresponding path length is not zero.
If motion is in straight line and in same direction, then the magnitude of displacement equals path length (distance).
If an object moving along the straight line covers equal distances in equal intervals of time, it is said to be in uniform motion.
There are two ways to determine the average velocity, numerically and graphically.
Numerically, average velocity is calculated as the net displacement (∆x) divided by total time (∆t). Direction of average velocity is same as direction of change in displacement (∆x).
On a position-time graph, the magnitude of average velocity between any two points is the slope of the straight line that connects these two particular points on x(t) curve. The slope of the straight line is tangent of the angle made by the line with the positive time-axis. A positive average velocity implies the line of slope slanting upward to the right. A negative average velocity implies the line of slope slanting downward to the right.
In kinematics, average speed is different than the average velocity.
Average speed is always positive, but average velocity can be positive and negative.
Average speed is always equal to or greater than the magnitude of the average velocity.
Zero Vector or null vector is a vector which has zero magnitude and an arbitrary direction.
Numerically, the instantaneous velocity is calculated by applying differential calculus as :
v = dx / dt
Magnitude of the instantaneous velocity |v| is the value of dx/dt and direction is same as direction of dx.
Graphically, a slope is drawn at specific point on position-time x(t) curve to the positive time-axis to find instantaneous velocity; whereas to find average velocity, a slope is drawn joining two points on position-time x(t) curve to the positive time-axis.
The value of slope of a line is tangent of angle made by this line with positive time-axis, which is the magnitude of velocity.
v = tan β = dx / dt
From position-time graph, direction of line’s slope indicates the direction of velocity. A positive velocity implies the line of slope slanting upward to the right. A negative velocity implies the line of slope slanting downward to the right.
Trigonometrically, the slope is positive in first and third quadrants, so velocity is positive; whereas the slope (velocity) is negative in second and fourth quadrants.
The derivative of position-time x(t) function is the velocity.
For uniform motion, instantons velocity (also called velocity) is the same as the average velocity at all instants. Moreover, in uniform motion, the position-time x(t) curve must be a straight line, velocity must be constant, and acceleration must be zero.
The first derivative of x(t) curve is velocity. The area under the v(t) curve is the change in displacement of the object.
Average velocity takes into account only the net change of displacement (final displacement – initial displacement) in an amount of time. The instantaneous velocity during this motion could be uniform or non-uniform.
Average acceleration is defined as the change in velocity vector ( Δv) divided by the time intervals (Δt).
Average acceleration is a vector quantity. Magnitude of average velocity is denoted as |<v>|. Direction of average acceleration < a > is same as direction of change in velocity ∆v.
Instantaneous acceleration (or simply acceleration) is the limit of average acceleration as the time interval Dt becomes infinitesimally small. We can also calculate acceleration by second derivative of position with respect to time.
Instantaneous acceleration is a vector quantity. Magnitude of instantaneous velocity is denoted as |<a> |. Direction of acceleration is same as direction of change in velocity dv.
Kinematic Equations for motion under constant acceleration:
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