To determine the kinetic energies of the objects and their order from least to greatest, we need to use the formula for kinetic energy:
[tex]\[ KE = \frac{1}{2} \times \text{mass} \times \text{velocity}^2 \][/tex]
Let's calculate the kinetic energy for each object.
For object [tex]\( W \)[/tex]:
[tex]\[ \text{mass}_W = 10 \text{ kg} \][/tex]
[tex]\[ \text{velocity}_W = 8 \text{ m/s} \][/tex]
[tex]\[ KE_W = \frac{1}{2} \times 10 \times 8^2 = \frac{1}{2} \times 10 \times 64 = 320 \text{ J} \][/tex]
For object [tex]\( X \)[/tex]:
[tex]\[ \text{mass}_X = 18 \text{ kg} \][/tex]
[tex]\[ \text{velocity}_X = 3 \text{ m/s} \][/tex]
[tex]\[ KE_X = \frac{1}{2} \times 18 \times 3^2 = \frac{1}{2} \times 18 \times 9 = 81 \text{ J} \][/tex]
For object [tex]\( Y \)[/tex]:
[tex]\[ \text{mass}_Y = 14 \text{ kg} \][/tex]
[tex]\[ \text{velocity}_Y = 6 \text{ m/s} \][/tex]
[tex]\[ KE_Y = \frac{1}{2} \times 14 \times 6^2 = \frac{1}{2} \times 14 \times 36 = 252 \text{ J} \][/tex]
For object [tex]\( Z \)[/tex]:
[tex]\[ \text{mass}_Z = 30 \text{ kg} \][/tex]
[tex]\[ \text{velocity}_Z = 4 \text{ m/s} \][/tex]
[tex]\[ KE_Z = \frac{1}{2} \times 30 \times 4^2 = \frac{1}{2} \times 30 \times 16 = 240 \text{ J} \][/tex]
Now, we need to order these kinetic energies from least to greatest:
[tex]\[ KE_X = 81 \text{ J} \][/tex]
[tex]\[ KE_Z = 240 \text{ J} \][/tex]
[tex]\[ KE_Y = 252 \text{ J} \][/tex]
[tex]\[ KE_W = 320 \text{ J} \][/tex]
Therefore, the order of the objects by their kinetic energy from least to greatest is:
[tex]\[ X, Z, Y, W \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{X , Z , Y , W} \][/tex]
