Analyzing the Velocity of a Skier on an Idealized Slope

The question of how long a skier would take to reach a specific speed down a slope is a common yet intriguing problem in the realm of physics. The scenario given states that a skier starts from rest and slides 9 meters down a slope and asks, 'In what time after starting from rest will the skier acquire a velocity of 24 m/s, assuming constant acceleration and negligible friction?' This problem requires careful analysis to determine the appropriate solution.

Assumptions and Key Variables

To solve this problem, several assumptions must be made. Firstly, it is assumed that the acceleration is constant and friction is negligible. These assumptions simplify the analysis but also limit the applicability of the solution to real-world scenarios.

Energy Conservation

Starting with the conservation of energy, we can use the relationship between kinetic energy (KE) and potential energy (PE) to understand the skier's motion. The energy conservation equation is:

Mechanical Energy (PE KE) Constant
Initial PE Initial KE Final PE Final KE

Since the skier starts from rest, the initial kinetic energy is 0. The equation simplifies to:

Mgmental Energy (PE KE) mg(h? - h)

As the skier moves down the slope, the potential energy decreases (mgh) and the kinetic energy increases (?mv2). Therefore, we have:

?mv2 mgh

Solving for h, we get:

h v2 / (2g)

Substituting the given values (v 24 m/s, g 9.8 m/s2):

h (24)2 / (2 × 9.8) 29.4 meters

This calculation shows that the skier needs to descend 29.4 meters vertically to reach a velocity of 24 m/s. However, the question mentions a descent of 9 meters along the slope. This discrepancy highlights the need to clarify the vertical distance versus the horizontal distance.

Uncertainty in the Problem

The problem statement is ambiguous regarding whether the 9 meters mentioned is the vertical or horizontal distance. To clarify, we need the angle of the slope (θ) to convert this distance between the two. The relationship is given by:

sinθ vertical distance / total distance along the slope

cosθ horizontal distance / total distance along the slope

Without this information, it is impossible to determine if the skier can reach the required velocity in the given distance.

Impact of Angle and Distance

The angle of the slope significantly affects the motion of the skier. If the slope is very shallow, the skier might not reach the desired speed due to the long horizontal distance required. Conversely, if the slope is steep, the skier could potentially reach the 24 m/s velocity faster.

Conclusion

In summary, the problem as stated does not provide sufficient information to determine the time required for the skier to reach 24 m/s after sliding 9 meters. To provide a valid solution, additional information such as the slope angle is necessary.

Assuming constant acceleration and ignoring friction, the time required cannot be determined directly from the given information. The vertical distance needed for the skier to achieve 24 m/s is 29.4 meters, which is greater than the 9 meters mentioned in the problem. Therefore, the skier would not reach 24 m/s after sliding only 9 meters down any slope.

Related Keywords

Skiing Physics, Velocity Calculation, Slope Angle, Kinetic Energy