Question

A man with a mass of \(65 \mathrm{kg}\) skis down a frictionless hill that is \(5.0 \mathrm{m}\) high. At the bottom of the hill the terrain levels out. As the man reaches the horizontal section, he grabs a \(20-\mathrm{kg}\) backpack and skis off a 2.0-m-high ledge. At what horizontal distance from the edge of the ledge does the man land?

Short answer

The man lands approximately 4.85 meters away from the edge of the ledge.

Step-by-step solution

Define the Energy Conservation Principle

When the man skis down the hill, the potential energy at the top is converted to kinetic energy at the bottom. This transformation assumes no friction, so the mechanical energy is conserved.

Calculate Potential Energy at the Top of the Hill

The potential energy (PE) at the top of the hill is given by \( PE = mgh \), where \( m = 65 \text{ kg} \), \( g = 9.8 \text{ m/s}^2 \), and \( h = 5.0 \text{ m} \). Therefore, \( PE = 65 \times 9.8 \times 5 = 3185 \text{ J} \).

Calculate Kinetic Energy at the Bottom of the Hill

Since energy is conserved, the kinetic energy (KE) at the bottom of the hill is equal to the potential energy at the top. Thus, \( KE = 3185 \text{ J} \).

Calculate Velocity at the Bottom of the Hill

The kinetic energy can also be expressed as \( KE = \frac{1}{2}mv^2 \), where \( v \) is the velocity. Solving for \( v \), we get \[ v = \sqrt{\frac{2 \times 3185}{65}} = \sqrt{98} \approx 9.9 \text{ m/s}. \]

Calculate Combined Mass Velocity After Grabbing Backpack

After grabbing the 20-kg backpack, the total mass is \( 65 + 20 = 85 \text{ kg} \). The conservation of momentum is applied: \( 65 \times 9.9 = 85 \times v_f \). Solving for \( v_f \), the new velocity, gives \[ v_f = \frac{65 \times 9.9}{85} \approx 7.57 \text{ m/s}. \]

Calculate Time of Flight from the Ledge

The time \( t \) it takes to fall 2 meters can be calculated using \( h = \frac{1}{2}gt^2 \). So:\( 2 = \frac{1}{2} \times 9.8 \times t^2 \) yields \[ t = \sqrt{\frac{2 \times 2}{9.8}} \approx 0.64 \text{ seconds}. \]

Determine the Horizontal Distance

The horizontal distance \( d \) is the horizontal velocity \( v_f \) multiplied by the time of flight \( t \). Therefore, \( d = 7.57 \times 0.64 \approx 4.85 \text{ meters}. \)

Key concepts

Energy Conservation

The principle of energy conservation means that energy cannot be created or destroyed, only transformed from one form to another. This is essential in many physics problems, including projectile motion. For the skier going down a hill, the energy conversion follows a path from potential energy, which he possesses at the top of the hill, to kinetic energy, which he gains as he descends. Since we are assuming a frictionless environment, no energy is lost to heat or other forms, allowing us to effectively track this transformation.

• At the top of the hill: Maximum potential energy, zero kinetic energy.
• At the bottom of the hill: Zero potential energy, maximum kinetic energy.

By understanding this transformation, we can calculate the speed of the skier at different points along his path, as energy values shift from potential to kinetic.

Kinetic Energy

Kinetic energy refers to the energy an object possesses due to its motion. This is calculated using the formula:\[ KE = \frac{1}{2}mv^2 \]Here, \( m \) is mass and \( v \) is velocity. In our scenario, the skier's potential energy at the top of the hill converts entirely to kinetic energy by the time he reaches the bottom, assuming no energy losses. Therefore, by knowing the total energy available, one can solve for velocity, which is a crucial factor for understanding motion.An increase in velocity will result in a greater kinetic energy since the kinetic energy is proportional to the square of the velocity. So even a small increase in speed can cause a notable increase in kinetic energy, making this concept vital for calculating how fast the skier will travel at varying stages of his descent.

Potential Energy

Potential energy is the stored energy of position possessed by an object. It is determined by the position of the object relative to a gravitational source. In our context, the potential energy of the skier is determined by his height above the ground at the top of the hill. The formula used is:\[ PE = mgh \]Where \( m \) is mass, \( g \) is acceleration due to gravity (\( 9.8 \text{ m/s}^2 \)), and \( h \) is height. The potential energy is at its maximum when the skier is at the top of the hill. As he moves downward, this potential energy is reduced, translating into kinetic energy.Understanding potential energy helps to predict how high the skier can travel when encountering other slopes or obstacles, providing insight into energy distribution and management throughout the motion.

Momentum Conservation

Momentum conservation is a fundamental principle in physics which states that the total momentum of a closed system is constant if no external forces act on it. When the skier grabs the backpack, the scenario illustrates momentum conservation vividly.The original momentum of the skier is maintained after grabbing the backpack, before skiing off the ledge. This means that:\[ m_1v_1 = (m_1 + m_2)v_f \] - \( m_1 \) is the initial mass (the skier) - \( v_1 \) is the initial velocity - \( m_2 \) is the mass of the backpack - \( v_f \) is the final velocity of the combined mass By applying this formula, the new velocity of the skier can be computed after the mass of the backpack is added, allowing us to predict travel after grabbing the backpack and moving off the ledge. Momentum conservation is crucial for understanding changes in velocity when objects interact or combine in motion.