Question

Think \& Calculate 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

The speed is less than \( 9.00 \, \text{m/s} \); it is \( 7.65 \, \text{m/s} \).

Step-by-step solution

Understand the Concept

The sled is moving without friction, implying that we can use the conservation of energy principle. Energy at the top of the hill (potential + initial kinetic) converts entirely into kinetic energy at the bottom (as potential energy becomes zero).

Establish Formula for First Run

In the first run, where the sled starts from rest, the potential energy at the top is completely converted into kinetic energy at the bottom. The kinetic energy at the bottom is given by \( KE = \frac{1}{2}mv^2 \), where \( v = 7.50 \, \text{m/s} \). This gives us \( mgh = \frac{1}{2}m(7.50)^2 \).

Establish Formula for Second Run

In the second run, the sled starts with an initial speed, so it has initial kinetic energy plus potential energy at the top. Using the same principle, \( mgh + \frac{1}{2}m(1.50)^2 = \frac{1}{2}mv^2 \), where \( v \) is the speed at the bottom for the second run.

Simplify and Compare

In both runs, the potential energy \( mgh \) cancels out, allowing us to solve for \( v \) in the second run using the equation from Step 3: \( \frac{1}{2}(7.50)^2 + \frac{1}{2}(1.50)^2 = \frac{1}{2}v^2 \). Solving for \( v \), we find \( v = \sqrt{(7.50)^2 + (1.50)^2} \).

Calculate the Result

Compute the speed in the second run: \( v = \sqrt{(7.50)^2 + (1.50)^2} = \sqrt{56.25 + 2.25} = \sqrt{58.5} \approx 7.65 \, \text{m/s} \).

Answer Part (a)

Since the computed speed \( 7.65 \, \text{m/s} \) is less than \( 9.00 \, \text{m/s} \), the speed at the bottom is less than \( 9.00 \, \text{m/s} \) after starting with an initial speed of \( 1.50 \, \text{m/s} \).

Finalize the Answer for Part (b)

The speed of the sled at the bottom of the hill in the second run is \( 7.65 \, \text{m/s} \).

Key concepts

Physics

Physics is the study of the natural world and the laws that govern it. In this scenario, we explore a critical physics principle known as the conservation of energy. The exercise involves a sled sliding down a frictionless hill, starting either from rest or with an initial speed. Now, how does this relate to physics? Well, physics helps us understand that when the sled slides down, it transforms energy from one type to another. Here, gravitational potential energy at the hill's top is converted into kinetic energy at the bottom, illustrating energy conservation. By doing so, we neglect frictional forces, meaning that no external force alters the sled's energy transformation.

The absence of friction lets us simplify the problem using the conservation of energy equation. All the potential energy from the top becomes kinetic energy at the bottom, showcasing a perfect conversion, which is a fundamental physics concept. So, physics is essentially showing us how different forms of energy are interchangeable without loss in an idealized environment.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, and it's calculated using the formula:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass and \( v \) is the velocity. In the exercise, the kinetic energy transforms significantly between the top and bottom of the hill.

Initially, when the sled is at the top, it may have some kinetic energy if it starts with an initial speed. As it moves down the hill, potential energy is converted to kinetic energy. At the bottom, all potential energy has become kinetic energy, thus increasing the sled's speed. This relationship highlights the kinetic energy's dependence on an object's velocity: as it speeds up, its kinetic energy increases.

For the sled starting from rest (first run), this kinetic energy at the bottom tells us how fast it travels. For the second run with a starting speed, this initial kinetic energy adds to what is gained from potential energy conversion, although still not enough to reach the speed of 9.00 m/s due to the energy constraints outlined earlier.

Potential Energy

Potential energy is the stored energy an object possesses because of its position, condition, or state. In the scenario given, it specifically refers to gravitational potential energy, which is dependent on:

• Height (\( h \))
• Mass of the sled (\( m \))
• Gravitational field strength (\( g \))

The formula is expressed as \( PE = mgh \). When the sled is at the top of the hill, this potential energy is at its maximum. As the sled slides down, this energy converts into kinetic energy.

In a frictionless scenario, this conversion is efficient, allowing potential energy to completely transform into kinetic energy. The greater the height, the more potential energy the sled has initially. Thus, a higher hill results in a greater speed at the bottom due to more energy being converted. This exercise demonstrates, in both runs, how potential energy at the top becomes kinetic energy at the bottom without any loss.

Frictionless Motion

Frictionless motion describes a scenario where no frictional forces act on moving objects. In this exercise, the sled slides down a hill without any friction. This idealization allows us to precisely apply the conservation of energy. Without friction, there's no energy loss as the sled moves, so calculations are simplified.

In reality, friction would slow down the sled, converting kinetic energy into heat, making it impossible to fully convert potential energy into kinetic energy. However, in frictionless motion, all potential energy transfers to kinetic energy without any deductions along the way. It's worth noting that setting up problems this way helps simplify complex real-world physics scenarios into more easily computable exercises.

Idealized frictionless motion enables us to use fundamental physics concepts to understand and predict motion outcomes, providing a clear example of how potential energy transforms into kinetic energy without interruptions.