Final answer:
Using the work-energy principle and the work done by gravity, the calculation indicates that the snowboarder travels down the ramp for approximately 39 meters. This solution assumes no other forces, such as friction, are at work besides gravity.
Step-by-step explanation:
To determine the distance the snowboarder traveled down the ramp, we can use the work-energy principle which states that the work done by all forces acting on an object equals the change in kinetic energy of the object. However, in this case, the snowboarder is descending, implying that the gravitational work done will be converted into kinetic energy and potential energy reduction. Since we are given the amount of work done by gravity (W = 2.6×104 J), we can use the equation W = F × d × cos(θ), where F is the force of gravity, d is the distance traveled down the slope, and θ is the angle of the incline with respect to the horizontal.
The force of gravity acting on the snowboarder, F, is equal to the snowboarder's mass, m, multiplied by the acceleration due to gravity, g (9.8 m/s2). Hence, F = m × g and for a 68 kg snowboarder F = 68 kg × 9.8 m/s2 = 666.4 N. Now, we can solve for distance d using Θ = 33° (below the horizontal).
The angle below horizontal doesn't affect F since it is along the incline. Thus, we should use θ = 0° in our work equation, which simplifies to W = F × d. Rearranging the equation to solve for d, we get d = W/F.
Substituting the known values, d = (2.6×104 J) / (666.4 N), which gives d approximately equal to 39 meters (since J = N × m).
This analysis assumes that all the work done by gravity is transferred to the snowboarder without losses, a scenario that would only happen in the absence of friction or other forces. Therefore, the snowboarder travels down 39 meters on the ramp, if only the force of gravity does work.