91Ó°ÊÓ

Gravitational force is the attractive force that pulls objects towards the center of the Earth. It acts on every object with mass and is given by Newton's law of gravitation. In this problem, the gravitational force acts on the snowball pulling it downward. The force of gravity can be calculated using the equation:

\( F = mg \)
where:

• \( m \) is the mass of the object (in kg),
• \( g \) is the acceleration due to gravity (9.8 m/s²).

For our snowball, the gravitational force pulling it downwards would be:

\( F = 1.50 \times 9.8 = 14.7 \text{ N} \)

This constant force is what causes the snowball to accelerate downwards and impacts the work done and energy calculations.

Potential energy is the energy stored in an object due to its position. Gravitational potential energy depends on the height of an object relative to some reference point. The formula to calculate it is:

\( U = mgh \)

Here:

• \( m \) is the mass of the object (1.50 kg),
• \( g \) is the acceleration due to gravity (9.8 m/s²),
• \( h \) is the height above the reference point (12.5 m in the case of the cliff).

The initial potential energy of the snowball at the top of the cliff can be calculated as:

\( U = 1.50 \times 9.8 \times 12.5 = 183.75 \text{ J} \)

This potential energy will change as the snowball falls.

Work done by gravity is the energy transferred to or from an object via the force of gravity as it moves. It can be calculated using the work-energy theorem, which states:

\( W = F \times d \times \text{cos(θ)} \)

In our scenario,

• The force (F) is the weight of the snowball (14.7 N),
• The displacement (d) is the vertical distance fallen (12.5 m),
• \(\theta\) is 0 degrees since the force and displacement are in the same direction.

This simplifies the work done by gravity to:

\( W = mgh = 1.50 \times 9.8 \times 12.5 = 183.75 \text{ J} \)

This means gravity does 183.75 J of work on the snowball as it falls.

Kinematics studies the motion of objects without considering the forces that cause this motion. In this problem, kinematics elements such as initial velocity and angle play a role. The initial velocity (14.0 m/s) and the angle (41.0 degrees) of the snowball's launch affect how it moves. The snowball's trajectory can be split into horizontal and vertical components using trigonometry:

• Vertical component: \( v_{y} = v \sin(\theta) = 14.0 \sin(41.0°) \)
• Horizontal component: \( v_{x} = v \cos(\theta) = 14.0 \cos(41.0°) \)

These components are crucial to calculate further details like maximum height reached or time of flight, but for our energy calculations, they primarily provide context for how the initial conditions influence the snowball's motion.

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In our scenario, the snowball's total mechanical energy (sum of kinetic and potential energy) remains constant if we neglect air resistance. Initially, the snowball has potential energy (due to its height) and kinetic energy (due to its velocity).

As it falls, potential energy is converted into kinetic energy. The conservation of energy equation can be written as:

\( U_i + K_i = U_f + K_f \)

where:

• \( U_i \) is initial potential energy,
• \( K_i \) is initial kinetic energy,
• \( U_f \) is final potential energy,
• \( K_f \) is final kinetic energy.

At ground level, the potential energy is zero if we take the reference height at ground level. The initial potential energy (183.75 J) has transformed into an equivalent amount of kinetic energy when the snowball hits the ground.