Problem 82 - College Physics

Question

\(\bullet\) \(\bullet\) 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 -kg skier with good form, friction and air resistance will do total work of magnitude 4000 J on him during his run down the slope. What is the max- imum height \(h\) for which the maximum safe speed will not be exceeded?

Step-by-Step Solution

Answer

Maximum height is approximately 50.37 meters.

Step 1: Understanding the Problem

We need to calculate the maximum vertical height \( h \) of a ski jump ramp from the starting gate to the bottom, ensuring the skiers do not exceed a speed of 30.0 m/s at the bottom. We will use the conservation of energy principle to solve this.

Step 2: List Known Values

The initial speed \( v_i = 2.0 \, \text{m/s} \), the final speed \( v_f = 30.0 \, \text{m/s} \), the mass of the skier \( m = 85.0 \, \text{kg} \), and the total work done by friction and air resistance \( W = 4000 \, \text{J} \). We need to find the height \( h \).

Step 3: Conservation of Energy Equation

The conservation of mechanical energy with friction considered can be stated as: \[\frac{1}{2} m v_i^2 + mgh = \frac{1}{2} m v_f^2 + W\]

Step 4: Plug in Known Values and Solve for h

Substitute the known values into the energy conservation equation:\[\frac{1}{2} \times 85.0 \times (2.0)^2 + 85.0 \times 9.8 \times h = \frac{1}{2} \times 85.0 \times (30.0)^2 + 4000\]Simplifying, we get:\[170 + 833h = 38250 + 4000\]\[833h = 42130 - 170\]\[833h = 41960\]Dividing by 833, we find:\[h = \frac{41960}{833} \approx 50.37 \, \text{m}\]

Step 5: Interpret the Result

The calculated height \( h \approx 50.37 \, \text{m} \) is the maximum vertical height that ensures skiers do not exceed a speed of 30.0 m/s at the bottom of the ramp.

Key Concepts

Kinetic Energy

Kinetic energy is an important concept to understand when designing a ski jump. It refers to the energy an object possesses due to its motion. For a skier, kinetic energy increases as they move faster down the slope.
This energy is calculated using the formula:\[KE = \frac{1}{2} mv^2\]

• \(m\) is the mass of the skier.
• \(v\) is the velocity.

This relationship means that as the skier's speed increases, their kinetic energy increases exponentially. At the top of the ski jump, the skier starts with a certain amount of kinetic energy, which then increases as they descend due to the conversion of potential energy into kinetic energy. Ensuring the safety of the skier involves managing this energy conversion to prevent them from exceeding a safe speed.

Potential Energy

Potential energy is the stored energy an object has because of its position or state. For the ski jump ramp, this refers to the height of the skier above the ground level.
This energy can be expressed as:\[PE = mgh\]

• \(m\) is the mass of the skier.
• \(g\) is the acceleration due to gravity, approximately \(9.8 \, \text{m/s}^2\).
• \(h\) is the height.

At the starting gate, skiers have a maximum amount of potential energy. As they descend the ramp, this potential energy decreases while being converted into kinetic energy.
The challenge in designing a ski jump ramp lies in balancing potential and kinetic energy to maintain safety while allowing the skier to achieve the desired speed.

Work-Energy Principle

The work-energy principle is a vital tool for understanding the energy transformations during the skier's descent. It states that the work done by all forces acting on an object equals the change in its kinetic energy.
During the ski jump:

• Skiers begin with a mix of potential and kinetic energy.
• Friction and air resistance perform work, dissipating some energy as heat.

The equation for the work-energy principle, including work done (\( W \)) by external forces such as friction, is:\[KE_i + PE_i + W = KE_f + PE_f\]By solving this equation, designers ensure energy is conserved and calculate parameters like the maximum height \( h \). This ensures the skier's final speed does not exceed safety limits.
This principle is essential for effectively applying conservation of energy in situations where other forces, like friction, are present.

Ski Jump Design

Designing a ski jump involves multiple physics principles to ensure athletes' safety and optimal performance. The primary considerations are managing the energy transformation and accounting for work done by friction and air resistance.
A successful ski jump design must achieve the following:

• Optimize height and slope to convert potential energy into kinetic energy efficiently.
• Limit friction and air resistance to manageable levels, as these can impede speed.
• Ensure the skier reaches the desired final speed, maximized by the calculated kinetic energy.

Incorporating these principles allows designers to calculate the crucial starting height \( h \) to ensure maximum safety without compromising the skier's speed. Effective design involves not only the height but also the slope's curvature and surface material, which impact how energy is conserved and converted throughout the jump.