Question

How much mechanical energy is lost to friction if a 55.0-kg skier slides down a ski slope at constant speed of $14.4\mathrm{m}/\mathrm{s}$ ? The slope is $123.5\mathrm{m}$ long and makes an angle of ${14.7}^{\circ }$ with respect to the horizontal.

Short answer

Answer: The mechanical energy lost to friction is approximately 41483.09 J.

Step-by-step solution

Calculate the vertical displacement of the skier.

To calculate the vertical displacement of the skier, we need to use the length of the slope ($123.5\mathrm{m}$) and the angle it makes with the horizontal (${14.7}^{\circ }$): $h=123.5\cdot sin({14.7}^{\circ })$ Using the sine function on our calculator, we get: $h\approx 31.48\mathrm{m}$

Determine the gravitational potential energy.

Now we can find the gravitational potential energy (GPE) when the skier is at the top of the slope. The GPE is given by the formula: $GPE=mgh$ By plugging in the known values, we get: $GPE=55.0\cdot 9.81\cdot 31.48\approx 17013.26\mathrm{J}$

Calculate the kinetic energy of the skier.

Since the skier slides down the slope at a constant speed ($14.4\mathrm{m}/\mathrm{s}$), we can find their kinetic energy (KE) using the following formula: $KE=\frac{1}{2}m{v}^2$ By plugging in the known values, we get: $KE=\frac{1}{2}\cdot 55.0\cdot (14.4{)}^2\approx 6471.04\mathrm{J}$

Calculate the total initial mechanical energy.

Now we can sum up the gravitational potential energy and the kinetic energy to find the initial total mechanical energy: $Total\mathrm{\_}ME=GPE+KE$ $Total\mathrm{\_}ME\approx 17013.26+6471.04\approx 23484.30\mathrm{J}$

Determine the work done by friction.

To find the work done by friction, we need to first find the normal force and then use the work-energy theorem. The normal force ($F_N$) can be found as follows: $F_N=mg\cdot cos({14.7}^{\circ })$ By plugging in the known values, we get: $F_N\approx 55.0\cdot 9.81\cdot cos({14.7}^{\circ })\approx 526.74\mathrm{N}$ Now we can use the work-energy theorem to find the work done by friction: $W_{friction}=F_{friction}\cdot d\cdot cos({180}^{\circ })$ Since the friction force is equal to the normal force, we have: $W_{friction}=526.74\cdot 123.5\cdot (-1)\approx -64967.39\mathrm{J}$

Calculate the mechanical energy lost to friction.

To find the mechanical energy lost to friction, we subtract the work done by friction from the initial total mechanical energy: $M{E}_{lost}=Total\mathrm{\_}ME-|W_{friction}|$ $M{E}_{lost}\approx 23484.30-64967.39\approx -41483.09\mathrm{J}$ The negative value indicates that the mechanical energy is indeed being lost to friction. Therefore, the mechanical energy lost to friction is approximately 41483.09 J.

Key concepts

Friction

Friction is a force that opposes motion between two surfaces that are in contact. In the case of our skier, friction acts as a resistance force between the skis and the snow surface.
This resistance consumes some of the skier's mechanical energy, converting it into heat.
Understanding how friction affects motion is key to solving problems where energy conversion is involved, like in this exercise.

• Friction operates opposite to the direction of motion.
• It is dependent on the normal force and the roughness of the surfaces in contact.
• Frictional force can be calculated by multiplying the normal force by the coefficient of friction.

In our exercise, friction plays a significant role because the skier moves at a constant speed, which implies that any gain in kinetic energy from sliding down is counteracted by the energy lost due to friction.

Gravitational Potential Energy

Gravitational potential energy (GPE) refers to the energy that an object possesses due to its position relative to the ground. It's higher if the object is held at a greater height.
In the skier's case, GPE is determined when the skier is at the top of the slope.

• The greater the height, the more gravitational potential energy.
• GPE can be calculated with the formula: $GPE=mgh$, where $m$ is mass, $g$ is the gravitational force (9.81 ${\text{m/s}}^2$ on Earth), and $h$ is the height.
• As the skier descends, GPE decreases while other forms of energy, like kinetic, might increase.

For this exercise, knowing the height from which the skier descends helps in calculating the initial energy content, vital for understanding how energy is transformed from one form to another.

Kinetic Energy

Kinetic energy (KE) is the energy an object has due to its motion.
It increases as the speed of the object increases.
For the skier, the kinetic energy remains constant since they descend the slope at a consistent speed throughout.

• KE depends on both the mass of the object and its velocity.
• The formula used to determine kinetic energy is $KE=\frac{1}{2}m{v}^2$.
• In scenarios where kinetic energy changes, it often corresponds with other friction forces or changes in potential energy.

Despite the frictional losses, this skier manages to maintain constant speed, showing how kinetic energy works in harmony with other energy forms.Understanding kinetic energy is crucial in appreciating the work-energy principle and its effects on moving objects.