Question

Starting from rest a 71.0−kg skier skis down a slope 8.00×10²m long that has an average incline of 40.0 ∘. The speed of the skier at the bottom of the slope is 20.2 m/s. How much work Wₚencemervative do nonconservative forces do on the skier?

Answer 1

The nonconservative forces do -48195 J of work on the skier. This is calculated using the work-energy principle, where the initial gravitational potential energy minus the final mechanical energy (sum of kinetic and potential energy) yields the work done. The negative sign indicates work done against the skier's motion.

Explanation:

The calculation of the work done by nonconservative forces on the skier involves applying the work-energy principle. Initially at rest, the skier's energy is solely gravitational potential energy, expressed as mgh, where m is the mass, g is the acceleration due to gravity, and h is the vertical height of the slope. As the skier descends the slope, potential energy is converted into kinetic energy, and at the bottom, the total mechanical energy is the sum of kinetic and potential energy.

The work done by nonconservative forces can be determined by taking the difference between the initial and final mechanical energies. In this scenario, the calculated work done by nonconservative forces is -48195 J, signifying energy loss during the descent.

The negative sign indicates that the work done is against the direction of the skier's motion. This energy loss can be attributed to factors such as friction, air resistance, and other dissipative forces acting on the skier as they navigate the slope. The numerical value of -48195 J represents the amount of work done by these nonconservative forces, resulting in a reduction of the skier's overall mechanical energy by this magnitude.

Answer 2

So, the work done by non-conservative forces on the skier is
`\(-14421.42 \, \text{J}\)`.

The work done by non-conservative forces can be determined using the work-energy principle, which states that the net work done on an object is equal to its change in kinetic energy. Mathematically, this can be expressed as:

`\[ W_{\text{net}} = \Delta KE \]`

The change in kinetic energy
`(\(\Delta KE\))` is the final kinetic energy
`(\(KE_{\text{final}}\))` minus the initial kinetic energy
`(\(KE_{\text{initial}}\))`.

`\[ \Delta KE = KE_{\text{final}} - KE_{\text{initial}} \]`

The kinetic energy of an object is given by the formula:

`\[ KE = (1)/(2) m v^2 \]`

where m is the mass of the object and v is its velocity.

1. Calculate the initial kinetic energy
`(\(KE_{\text{initial}}\))` when the skier starts from rest:

`\[ KE_{\text{initial}} = (1)/(2) m v_{\text{initial}}^2 \]`

Since the skier starts from rest,
`\( v_{\text{initial}} = 0 \)`.

2. Calculate the final kinetic energy
`(\(KE_{\text{final}}\))` when the skier reaches the bottom of the slope:

`\[ KE_{\text{final}} = (1)/(2) m v_{\text{final}}^2 \]`

3. Calculate the change in kinetic energy
`(\(\Delta KE\))`:

`\[ \Delta KE = KE_{\text{final}} - KE_{\text{initial}} \]`

4. The work done by non-conservative forces
`(\(W_{\text{non-conservative}}\))` is equal to
`\(-\Delta KE\)` (negative because non-conservative forces typically do negative work):

`\[ W_{\text{non-conservative}} = -\Delta KE \]`

Now, let's substitute the given values:

`\( m = 71.0 \, \text{kg} \)`

`\( v_{\text{initial}} = 0 \, \text{m/s} \)` (starting from rest)

`\( v_{\text{final}} = 20.2 \, \text{m/s} \)`

Calculate
`\( KE_{\text{initial}} \), \( KE_{\text{final}} \), and then find \( W_{\text{non-conservative}} \)`.

1. Calculate
`\( KE_{\text{initial}} \)`:

`\[ KE_{\text{initial}} = (1)/(2) m v_{\text{initial}}^2 \]`

Since the skier starts from rest
`(\( v_{\text{initial}} = 0 \))`,

`\[ KE_{\text{initial}} = (1)/(2) * 71.0 \, \text{kg} * (0 \, \text{m/s})^2 = 0 \, \text{J} \]`

2. Calculate
`\( KE_{\text{final}} \)`:

`\[ KE_{\text{final}} = (1)/(2) m v_{\text{final}}^2 \]`

`\[ KE_{\text{final}} = (1)/(2) * 71.0 \, \text{kg} * (20.2 \, \text{m/s})^2 = 14421.42 \, \text{J} \]`

3. Calculate
`\( \Delta KE \)`:

`\[ \Delta KE = KE_{\text{final}} - KE_{\text{initial}} = 14421.42 \, \text{J} - 0 \, \text{J} = 14421.42 \, \text{J} \]`

4. Calculate
`\( W_{\text{non-conservative}} \)`:

`\[ W_{\text{non-conservative}} = -\Delta KE = -14421.42 \, \text{J} \]`