Problem 21 - College Physics

Question

\(\cdot\) \(\cdot\) You throw a 20 \(\mathrm{N}\) rock into the air from ground level and observe that, when it is 15.0 \(\mathrm{m}\) high, it is traveling upward at 25.0 \(\mathrm{m} / \mathrm{s} .\) Use the work-energy principle to find (a) the rock's speed just as it left the ground and (b) the maximum height the rock will reach.

Step-by-Step Solution

Answer

Initial speed is 28.4 m/s; Maximum height is 41.2 m.

Step 1: Understand the Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in its kinetic and potential energy. This can be expressed as: \( W = \Delta KE + \Delta PE \). We need to find the initial speed of the rock just as it leaves the ground and the maximum height it will reach using this principle.

Step 2: Identify Given Values and Equations

We know the rock's weight is 20 N, which means its mass \( m = \frac{20}{9.8} \approx 2.04 \text{ kg} \). At a height of 15.0 m, its velocity \( v = 25.0 \text{ m/s} \). We can use the equations \( KE = \frac{1}{2}mv^2 \) and \( PE = mgh \) to calculate the energies. Gravitational potential energy is \( P.E. = mgh \) and kinetic energy is \( K.E. = \frac{1}{2}mv^2 \).

Step 3: Calculate Initial Speed Using Energy Principles

Initially, the rock has no potential energy (it's on the ground). The total energy at 15 m is \( KE + PE = \frac{1}{2}mv^2 + mgh \). Solve for initial kinetic energy and use \( v^2 = u^2 + 2as \) (where \( u \) is initial speed) to find \( u = \sqrt{v^2 + 2gh} \). \( u = \sqrt{(25)^2 + 2\times9.8\times15} \approx 28.4 \text{ m/s} \).

Step 4: Calculate Maximum Height Using Energy Principles

When the rock reaches its maximum height, all its kinetic energy is transformed into potential energy: \( mgh_{max} = \frac{1}{2}mu^2 \). Solve for maximum height \( h_{max} \): \( h_{max} = \frac{u^2}{2g} = \frac{(28.4)^2}{2\times9.8} \approx 41.2 \text{ m} \).

Step 5: Conclusion

The solution is checked for consistent units and logical results. The conservation of energy has been used to solve for initial speed and maximum height correctly.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It depends on the mass and velocity of the object. The formula for calculating kinetic energy (KE) is given by:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass of the object and \( v \) is its velocity. In our example, as the rock is launched into the air, it possesses a certain amount of kinetic energy, which we calculated using its known velocity of 25.0 m/s at a height of 15.0 m.
The concept of kinetic energy helps us understand how the rock's speed affects the energy it carries. The faster the rock, the more kinetic energy it has. This energy changes as the rock moves, which links directly to how energy transfers between kinetic and potential forms.

Potential Energy

Potential energy is stored energy based on an object's position relative to other objects. For gravitational potential energy, this is a function of height and gravity. The formula for potential energy (PE) is:

• \( PE = mgh \)

where \( m \) is mass, \( g \) is the acceleration due to gravity (9.8 m/s² on Earth), and \( h \) is the height above the ground.
In our problem, as the rock rises 15 meters into the air, it gains potential energy. This is because it has moved against the force of gravity. Gravitational potential energy tells us how high the rock can go based on the energy available. When initially released from the ground, it has no potential energy until it starts climbing upwards.

Conservation of Energy

The conservation of energy is a principle stating that energy cannot be created or destroyed, only transformed from one form to another. In the case of the rock, mechanical energy—kinetic and potential combined—remains constant as it travels. According to this law, the total energy at any point during the rock's flight is the same as at launch.
We use this principle to solve the problem by equating the initial kinetic energy (when potential energy is zero at the ground) with the total energy at 15 meters (kinetic plus potential energy). This same total allows us to calculate the rock's speed at the launch and determine the highest point it can reach, known as maximum height.

Gravitational Forces

Gravitational force is an attractive force that pulls two masses toward each other, in this case, the Earth and the rock. The strength of this force is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.
For the rock, gravitational force is calculated as its weight multiplied by gravity (20 N), which allows us to determine its mass using the relation \( m = \frac{weight}{g} \). In our scenario, gravity acts upon the rock to bring it back towards Earth as it ascends. This force opposes the rock's motion, converting kinetic energy into potential energy as it climbs.

• It is essential to understand this force as it guides calculations related to energy transformations.
• Gravitational forces ensure that the potential energy at maximum height is not lost but reconverted to kinetic energy when it descends.

The comprehension of gravitational forces allows students to grasp how height affects the rock's potential to do work through energy changes.