Problem 56 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A meteorite has a speed of \(90.0 \mathrm{~m} / \mathrm{s}\) when \(850 \mathrm{~km}\) above the Earth. It is falling vertically (ignore air resistance) and strikes a bed of sand in which it is brought to rest in \(3.25 \mathrm{~m}\). (a) What is its speed just before striking the sand? (b) How much work does the sand do to stop the meteorite (mass \(=575 \mathrm{~kg}\) )? (c) What is the average force exerted by the sand on the meteorite? (d) How much thermal energy is produced?

Step-by-Step Solution

Answer

Approximate speed before impact: 4091 m/s. Work done by sand: -4,790,878,878.75 J. Average force by sand: -1474168796.38 N. Thermal energy: 4,790,878,878.75 J.

Step 1: Determine the Final Speed Before Impact

The initial speed of the meteorite when it's 850 km above Earth is 90.0 m/s. We use the formula for gravitational potential energy to find the speed just before striking the sand. The conversion from height to velocity can be calculated using energy conservation. The change in potential energy is converted to kinetic energy:\[ v^2 = u^2 + 2gs \]where \( u = 90 \text{ m/s} \), \( s = 850,000 \text{ m} \), \( g = 9.81 \text{ m/s}^2 \).Now substitute the values:\[ v^2 = 90^2 + 2(9.81)(850,000) \]Calculate \( v \):\[ v = \sqrt{90^2 + 2 \times 9.81 \times 850,000} \approx 4091 \text{ m/s} \].Thus, the speed just before striking the sand is approximately 4091 m/s.

Step 2: Calculate the Work Done by the Sand

The work done by the sand to stop the meteorite can be found using the work-energy principle. The work done is equal to the change in kinetic energy of the meteorite.\[ \text{Work} = \frac{1}{2} m v^2 - \frac{1}{2} m u^2 \]where \( m = 575 \text{ kg} \), \( v = 0 \text{ m/s} \), \( u = 4091 \text{ m/s} \).Substitute the known values:\[ \text{Work} = \frac{1}{2} \times 575 \times 4091^2 - 0 \]Calculate the work done:\[ \text{Work} = 0 - \frac{1}{2} \times 575 \times 4091^2 = -4790878878.75 \text{ J} \].The sand does \(-4,790,878,878.75 \text{ J}\) of work to stop the meteorite.

Step 3: Determine the Average Force Exerted by the Sand

The average force exerted by the sand can be calculated using the work done and the distance over which the force acts.\[ \text{Force} = \frac{\text{Work}}{\text{distance}} \]where the distance is \(3.25 \text{ m} \) and the work is \(-4790878878.75 \text{ J}\).\[ \text{Force} = \frac{-4790878878.75}{3.25} \]Calculate the force:\[ \text{Force} \approx -1474168796.38 \text{ N} \].The average force is approximately \(-1474168796.38 \text{ N}\) (the negative sign indicates direction).

Step 4: Calculate the Thermal Energy Produced

The thermal energy produced is equal to the work done by the sand as all the kinetic energy is converted into thermal energy.The thermal energy produced is \( 4,790,878,878.75 \text{ J} \), which is the magnitude of the work done to bring the meteorite to rest.

Key Concepts

Kinetic Energy

Understanding kinetic energy is crucial to solving problems related to moving objects. Kinetic energy is the energy an object possesses due to its motion. It is given by the formula \[ KE = \frac{1}{2}mv^2 \] where \( m \) is the mass of the object, and \( v \) is the velocity.

• The greater the speed, the higher the kinetic energy.
• Kinetic energy depends on the square of the velocity, which means doubling the speed quadruples the kinetic energy.

In the exercise, the meteorite's kinetic energy changes twice. First, as it accelerates towards Earth due to gravity. Then, when the kinetic energy is converted to other forms of energy as the meteorite is stopped by the sand.
Recognizing changes in kinetic energy helps in understanding how energies transform as the meteorite travels and eventually comes to rest.

Potential Energy

Potential energy is the stored energy based on position, height, or configuration. For objects near the Earth’s surface, gravitational potential energy can be calculated using:\[ PE = mgh \]where \( m \) is mass, \( g \) is gravity (\( 9.81 \, \text{m/s}^2 \)), and \( h \) is height above the ground.

• Potential energy is higher when the object is higher above the ground.
• Potential energy converts to kinetic energy as the object falls.

In the exercise, the meteorite has significant potential energy at 850 km above the Earth, which decreases as it falls, increasing its kinetic energy instead.
This conversion plays a critical role in determining the speed just before impact.

Average Force

The average force relates to how energy and movement are controlled over a particular distance or time. It is calculated using the formula:\[ F_{\text{avg}} = \frac{\text{Work done}}{\text{Distance}} \]

• The force is the measure of how much the sand slowed the meteorite down over 3.25 meters.
• A greater force implies more work was done over a shorter distance.

In this context, the negative sign of force suggests that the direction of force exerted by the sand is opposite to the meteorite’s motion.
Understanding average force provides insight into how quickly and efficiently a force can bring an object to rest. Force is essential in understanding impacts and collisions.

Thermal Energy

Thermal energy arises from the conversion of kinetic energy when objects and surfaces interact by contact. In the exercise, this is the energy generated as the meteorite comes to halt due to the sand's resistance.

• All kinetic energy of the meteorite converts to thermal energy since it's brought to a stop.
• Thermal energy is seen in the form of heat which may increase the temperature of both the meteorite and sand slightly.

This conversion is complete because the work done by the sand, through stopping the meteor, completely dissipates the initial kinetic energy into heat. It provides insight into energy transformation during interactions between surfaces.