"If a particle is speeding up, acceleration is in the direction of velocity. If its speed is decreasing, acceleration is the direction opposite to that of the velocity. This statement is independent of the choice of the origin and the axis." Can you please explain the above (with the help of a diagram perhaps)?
The illustration shows a velocity time graph. Initially the velocity of the object increases and then decreases to zero.
Considering the point A and B. The corresponding velocities are v1 and v2 and v2 > v1.
The acceleration is,
a = (v2 - v1)/(t2 – t1)
Which is a positive quantity. The positive value of acceleration implies that the acceleration and the velocity have the same direction.
Considering the point C and D. The corresponding velocities are v3 and v4 and v3 > v4.
The acceleration is,
a = (v4 – v3)/(t4 – t3)
Which is a negative quantity. The negative value of acceleration implies that the acceleration and the velocity have opposite direction.