Question

A sled slides without friction down a small, ice-covered hill. If the sled starts from rest at the top of the hill, its speed at the bottom is \(7.50 \mathrm{~m} / \mathrm{s}\). (a) On a second run, the sled starts with a speed of \(1.50 \mathrm{~m} / \mathrm{s}\) at the top. When it reaches the bottom of the hill, is its speed \(9.00 \mathrm{~m} / \mathrm{s}\), more than \(9.00 \mathrm{~m} / \mathrm{s}\), or less than \(9.00 \mathrm{~m} / \mathrm{s}\) ? Explain. (b) Find the speed of the sled at the bottom of the hill after the second run.

Short answer

(a) Less than 9.00 m/s. (b) 7.65 m/s.

Step-by-step solution

Understanding the Energy Conservation Principle

When the sled slides down a frictionless hill, mechanical energy is conserved. This means that the total mechanical energy at the top of the hill (potential energy + kinetic energy) is equal to the total mechanical energy at the bottom of the hill.

Calculating Initial and Final Energies for the First Run

On the first run, the sled begins at rest, so its initial kinetic energy is 0. The speed at the bottom is given as \(7.50 \, \mathrm{m/s}\). Thus, its kinetic energy at the bottom is \( \frac{1}{2}mv^2 \), where \( v = 7.50 \, \mathrm{m/s}\).

Relating Energies for the First Run

Because mechanical energy is conserved, the initial potential energy at the top is equal to the kinetic energy at the bottom of the hill. Calculate the initial potential energy: \( PE_{initial} = \frac{1}{2}m(7.50)^2 \).

Examining the Second Run Initial Energies

For the second run, the sled starts with an initial speed of \(1.50 \, \mathrm{m/s}\), so it has initial kinetic energy at the top as well as potential energy. The total initial energy at the top is \( KE_{initial} + PE = \frac{1}{2}m(1.50)^2 + PE \). Since the hill's height is the same, the corresponding potential energy is equal to the initial potential energy from the first run.

Calculating the Final Speed in the Second Run

Because energy conservation holds, the total energy at the top equals the kinetic energy at the bottom: \[ \frac{1}{2}m(1.50)^2 + PE = \frac{1}{2}m(v_{bottom})^2 \]. From this equation and the fact that \( PE \) from both runs is the same, calculate \( v_{bottom} \), finding that \[ (1.50)^2 + 2PE/m = v_{bottom}^2 \]. Since \( 2PE/m = (7.50)^2 \), the equation simplifies to \[ (1.50)^2 + (7.50)^2 = v_{bottom}^2 \].

Final Calculation Steps

Perform the final calculation: \[ 1.50^2 = 2.25 \] and \[ 7.50^2 = 56.25 \], so \[ v_{bottom}^2 = 2.25 + 56.25 = 58.50 \]. Take the square root to find \( v_{bottom} \): \[ v_{bottom} = \sqrt{58.50} \approx 7.65 \, \mathrm{m/s} \].

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is given by the equation \[ \text{KE} = \frac{1}{2} mv^2 \] where \( m \) is the mass of the object and \( v \) is its velocity. In the context of the sled problem, when the sled descends the hill, its speed increases, thereby increasing its kinetic energy.

• During the first run, the sled starts from rest, so its initial kinetic energy is zero.
• As the sled moves down the hill, it gains kinetic energy, reaching a maximum at the bottom, dictated by its velocity of \( 7.50 \mathrm{~m/s} \).

This gain in kinetic energy during the descent is a key component of understanding how mechanical energy converts from one form to another.

Potential Energy

Potential energy is the stored energy of an object due to its position in a field, often gravitational in simple mechanics problems. For an object near the Earth’s surface, gravitational potential energy is given by\[ \text{PE} = mgh \] where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height of the object above a reference point.
In our sled problem, the top of the hill represents a point of maximum potential energy since this is where the sled is highest.

• The first run begins with maximum potential energy, which transforms into kinetic energy as the sled descends.
• In the second run, the sled starts with both kinetic and potential energy, this combined energy equals the total energy at the hill's base.

This exchange from potential to kinetic energy is fundamental in understanding the principle of energy conservation in this scenario.

Mechanical Energy

Mechanical energy is the sum of an object's kinetic and potential energy. It's a conserved quantity in systems where only conservative forces, like gravity, act. For the sled scenario: - At the top of the hill, the sled's mechanical energy is all potential energy during the first run, because the kinetic energy is zero. - In the second run, this mechanical energy at the top includes the initial kinetic energy due to the sled's non-zero starting speed. Maintaining mechanical energy in this way showcases the principle that, despite different starting conditions, the total mechanical energy remains equal between top and bottom locations of the hill, giving rise to energy transformations between potential and kinetic forms without loss.

Frictionless Surfaces

Frictionless surfaces are idealized surfaces where no energy is lost to friction. This means all mechanical energy in a system is conserved and any change in motion only results from conversions between kinetic and potential energy, not energy dissipation. In our sled problem, the ice-covered hill is considered a frictionless surface. This ensures:

• The sled experiences no energy loss to friction as it descends.
• Any change in the sled's speed is solely due to conversions between potential and kinetic energy.

Such conditions simplify calculations and help illustrate core energy conservation principles by removing complicating factors like heat dissipation typically associated with friction.