Final answer:
The distance a cart travels down a 6-degree slope in 12 seconds without friction is determined by using the equation d = ut + ½at², taking into account the acceleration component of gravity along the slope.
Step-by-step explanation:
The question asks about the distance traveled by a cart rolling down a slope at a 6-degree angle to the horizontal over a period of 12 seconds without friction. To solve this, we can use the equations of motion under constant acceleration. Since there is no friction, the only force acting on the cart along the slope is gravity. The component of gravitational acceleration along the slope, g sin(θ), where θ is the angle of the slope, will be the acceleration of the cart. The gravitational acceleration, g, is approximately 9.81 m/s². The distance traveled can then be calculated using the formula d = ut + ½at² where u is the initial velocity (zero if starting from rest), a is the acceleration, and t is the time.
For the given situation:
- Initial velocity u = 0 m/s (since it starts from rest)
- Acceleration a = g sin(θ) = 9.81 m/s² * sin(6 degrees)
- Time t = 12 seconds
By substituting these values into the distance formula, we get:
d = 0 * 12 + ½ * 9.81 * sin(6 degrees) * (12)²
After calculating this, we find the distance d that the cart travels down the slope in 12 seconds.