A ball is travelling with uniform translatory motion. This means that

The correct option is: all parts of the ball have the same velocity (magnitude and direction) and the velocity is constant

Step-by-Step Explanation:

1. Understanding Translatory Motion:
Translatory motion (or translational motion) is a type of motion in which all points of a moving body move uniformly along parallel paths. This means that every single particle or part of the body undergoes the same displacement in the same interval of time.

2. Analyzing the Velocity of the Body's Parts:
Because every point of the ball experiences the exact same displacement over any given time interval, the velocity vectors of all parts of the ball must be identical. Therefore, at any instant, all parts of the ball have the same velocity, which includes both the same magnitude (speed) and the same direction.

3. Applying the "Uniform" Constraint:
The question specifies that the ball is undergoing uniform translatory motion. The term "uniform" means that this motion does not change over time. Thus, the velocity remains constant throughout the motion.

4. Evaluating the Options:
- "it is at rest": Incorrect, as a body at rest is not in motion.
- "the path can be a straight line or circular and the ball travels with uniform speed": Incorrect, because in circular motion, the direction of velocity constantly changes, meaning the velocity is not constant in direction, and it would involve rotational/centripetal components.
- "the centre of the ball moves with constant velocity and the ball spins about its centre uniformly": Incorrect, as spinning (rotation) means different parts of the ball have different velocities (speed and direction vary with distance from the axis of rotation). This represents a combination of translation and rotation, not pure translatory motion.
- "all parts of the ball have the same velocity (magnitude and direction) and the velocity is constant": Correct, as this perfectly matches the definition of uniform translatory motion.