Problem 52 - University Physics with Modern Physics

Question

Ski Jump Ramp. You are designing a ski jump ramp for the next Winter Olympics. You need to calculate the vertical height \(h\) from the starting gate to the bottom of the ramp. The skiers push off hard with their ski poles at the start, just above the starting gate, so they typically have a speed of 2.0 \(\mathrm{m} / \mathrm{s}\) as they reach the gate. For safety, the skiers should have a speed of no more than 30.0 \(\mathrm{m} / \mathrm{s}\) when they reach the bottom of the ramp. You determine that for a \(85.0-\mathrm{kg}\) skier with good form, friction and air resistance will do total work of magnitude 4000 \(\mathrm{J}\) on him during his run down the slope. What is the maximum height \(h\) for which the maximum safe speed will not be exceeded?

Step-by-Step Solution

Answer

The maximum height is approximately 50.5 meters.

Step 1: Write down the given data

Initial speed of the skier, \(v_i = 2.0 \, \text{m/s}\). Final speed at the bottom, \(v_f = 30.0 \, \text{m/s}\). Mass of the skier, \(m = 85.0 \, \text{kg}\). Work done against friction and air resistance, \(W = 4000 \, \text{J}\). We need to find the vertical height \(h\).

Step 2: Apply conservation of energy principle

The mechanical energy at the starting point plus any work done by non-conservative forces (like friction) should equal the mechanical energy at the bottom of the ramp. The equation is:\[KE_i + PE_i + W = KE_f + PE_f\]Where \(KE_i\) is the initial kinetic energy, \(PE_i\) is the initial potential energy, \(KE_f\) is the final kinetic energy, and \(PE_f\) is the final potential energy.

Step 3: Express kinetic and potential energies

The initial kinetic energy \(KE_i = \frac{1}{2}mv_i^2\), potential energy at the top \(PE_i = mgh\), final kinetic energy \(KE_f = \frac{1}{2}mv_f^2\), and potential energy at the bottom \(PE_f = 0\) since all potential energy has been converted to kinetic energy or lost to work. Substitute and simplify:\[\frac{1}{2}mv_i^2 + mgh - W = \frac{1}{2}mv_f^2\]

Step 4: Solve for the height h

Substitute the given values into the equation:\[\frac{1}{2}(85)(2.0)^2 + 85 \cdot 9.8 \cdot h - 4000 = \frac{1}{2}(85)(30.0)^2\]Simplifying this:\[0.5(85)(4) + 85(9.8)h - 4000 = 0.5(85)(900)\]\[170 + 833h - 4000 = 38250\]\[833h = 38250 - 170 + 4000\]\[833h = 42080\]\[h = \frac{42080}{833} \approx 50.5 \, \text{m}\]

Step 5: State the result

The maximum height \(h\) for which the skiers can safely reach the bottom of the ramp without exceeding the speed limit is approximately 50.5 meters.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It depends on two main factors: the mass of the object and its velocity. In mathematical terms, kinetic energy (\( KE \)) can be expressed as:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass and \( v \) is the velocity. For a moving skier, kinetic energy changes as they move down the slope reflecting the skier's speed changes.
At the beginning of a ski jump, the skier has an initial kinetic energy derived from their initial push-off speed. This initial kinetic energy at the gate can be calculated using the skier's mass and their initial speed of 2.0 \( \text{m/s} \). As the skier approaches the bottom with a higher speed (up to 30.0 \( \text{m/s} \)), their kinetic energy increases.
Ultimately, kinetic energy helps us understand how fast the skier will be moving at any given point on the jump on the ramp.

Potential Energy

Potential energy is the stored energy of an object due to its position. In the context of ski jumps, it's particularly associated with gravitational potential energy. This energy is determined by the height of the skier above a reference point, such as the bottom of the ramp. Mathematically, potential energy (\( PE \)) is expressed as:

• \( PE = mgh \)

where \( m \) is mass, \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)), and \( h \) is the height.
For the skier at the top of the ramp, this energy is at its maximum. As they descend, potential energy is converted into kinetic energy.
This conversion helps achieve the maximum speed at the bottom of the ramp, showcasing the fundamental energy transformations occurring during the ski jump.

Work-Energy Principle

The work-energy principle is essential in analyzing the movement of skiers in a jump. It states that the work done by the forces acting on an object is equal to the change in the object's kinetic energy. It's articulated as:

• Work done + Initial Mechanical Energy = Final Mechanical Energy

In the context of the ski jump, non-conservative forces such as friction and air resistance do work on the skier, altering their energy state.
In this exercise, 4000 \( \text{J} \) of energy is spent overcoming these forces. The initial mechanical energy (sum of initial kinetic and potential energy) decreases by this amount, influencing the skier's speed as they reach the bottom.
By understanding this principle, we can calculate how much potential energy needs to be present at the top to account for energy lost due to work done by friction and maintain safety speed limits.

Ski Jump Physics

Ski jump physics combines various principles of mechanics to ensure skiers safely navigate the ramp and achieve optimal performance. Central to this is energy conservation, where potential energy at the top of the ramp becomes kinetic energy at the bottom. The skier's design speed is carefully calculated to ensure safe descent.
The maximum speed attainable at the bottom of the ramp must not exceed 30.0 \( \text{m/s} \) for safety. This involves a delicate balance between initial kinetic energy, potential energy at the top, and energy work done against friction.
Calculating the ramp's height is crucial as it determines the initial potential energy available for conversion. By precisely estimating this height, we ensure the ski jump works within physics laws, making ski jumping both exciting and safe.
Overall, ski jump physics is a practical application of energy principles, demonstrating how conservation laws guide sports safety and performance.