Final answer:
The average velocity component squared in the x-direction of a ball with a known kinetic energy is independent of direction, and the average value is calculated to be 2.30 m/s.
Step-by-step explanation:
The question given pertains to the kinetic energy of a moving ball and its average velocity component in the x-direction. Since the average kinetic energy of the ball is known to be 2.30 J and the mass of the ball is 42.0 g, we can use the formula for kinetic energy (K = 1/2mv²) to find the average value of the velocity component squared (v²). Because velocity is a vector quantity, its square and thereby the kinetic energy are independent of the direction. Therefore, we can say that the average velocity component in the x-direction is the same as in any other direction and its determination doesn't depend on the direction of the x-axis.
By rearranging the kinetic energy formula, we get v² = 2K/m, which simplifies to the square of the average velocity component in the x-direction (v₂ₓ). Inserting the given values (converted to appropriate SI units), we find that the average value of v₂ₓ is indeed 2.30 m/s and it doesn't matter which direction is chosen to be x, answering part (a) of the original question with option 'b' (2.30 m/s, no).