A 5.0 kg wooden sled is launched up a 25° snow-covered slope with an initial speed of 10 m/s. What vertical height does the sled reach above its starting point?

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Step 1: Identify the forces acting on the sled. The sled is moving up the slope, so it is subject to gravitational force, which has a component acting down the slope. The initial kinetic energy of the sled will be converted into potential energy as it climbs the slope.

Step 2: Use the conservation of energy principle. The initial kinetic energy of the sled is given by \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the sled and \( v \) is its initial velocity. The potential energy at the maximum height is given by \( PE = m g h \), where \( g \) is the acceleration due to gravity and \( h \) is the vertical height.

Step 3: Set up the energy conservation equation. At the maximum height, all the initial kinetic energy is converted into potential energy: \( \frac{1}{2} m v^2 = m g h \). Simplify this equation to solve for \( h \): \( h = \frac{v^2}{2 g} \).

Step 4: Account for the slope angle. The vertical height \( h \) is related to the distance traveled along the slope \( d \) by the angle \( \theta \): \( h = d \sin \theta \). Use trigonometry to ensure the vertical height is calculated correctly.

Step 5: Combine the equations. Substitute the values for \( m = 5.0 \; \text{kg} \), \( v = 10 \; \text{m/s} \), \( g = 9.8 \; \text{m/s}^2 \), and \( \theta = 25° \) into the equations to calculate the vertical height \( h \). Ensure units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v², where m is mass and v is velocity. In this scenario, the sled's initial kinetic energy will be converted into potential energy as it moves up the slope.

Potential Energy

Potential energy is the energy stored in an object due to its position in a gravitational field, given by the formula PE = m * g * h, where m is mass, g is the acceleration due to gravity, and h is height. As the sled ascends the slope, its kinetic energy is transformed into potential energy until it reaches its maximum height.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this problem, the total mechanical energy (kinetic plus potential) of the sled remains constant as it moves up the slope, allowing us to equate the initial kinetic energy to the potential energy at the maximum height.

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