Final answer:
The final speed of the cart after a 68 kg rock is dropped into it is calculated using the conservation of momentum. The provided options, however, do not match the calculated value of approximately 21.1 m/s, suggesting a possible error in the question or options.
Step-by-step explanation:
The subject of the question involves calculating the final speed of a cart after a rock is dropped into it, which is a physics problem related to conservation of momentum. To solve this, we can use the principle that the total momentum before an object is dropped into the cart must equal the total momentum after the object is added, assuming no external forces act on the system.
Initial momentum of the cart (pi,cart) is the mass of the cart multiplied by its velocity: pi,cart = 500 kg * 24 m/s = 12000 kg*m/s. When the 68 kg rock is dropped into the cart, the total mass becomes 568 kg. The final velocity (vf) can be found by dividing the initial total momentum by the final total mass: vf = 12000 kg*m/s / 568 kg.
Performing the calculation gives us vf ≈ 21.1 m/s, which doesn't match any of the provided options. It seems like there may have been a mistake in the question or the possible responses since none of the options is close to the calculated value.