Problem 53 - University Physics with Modern Physics

Question

The Great Sandini is a \(60-\mathrm{kg}\) circus performer who is shot from a cannon (actually a spring gun). You don't find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1100 \(\mathrm{N} / \mathrm{m}\) that he will compress with a force of 4400 \(\mathrm{N}\) . The inside of the gun barrel is coated with Teflon, so the average friction force will be only 40 \(\mathrm{N}\) during the 4.0 \(\mathrm{m}\) he moves in the barrel. At what speed will he emerge from the end of the barrel, 2.5 \(\mathrm{m}\) above his initial rest position?

Step-by-Step Solution

Answer

15.5 m/s

Step 1: Calculate the compression of the spring

The force used to compress the spring is equal to the spring constant multiplied by the distance compressed, given by the formula: \[ F = kx \]Rearranging for compression distance \( x \), we have: \[ x = \frac{F}{k} = \frac{4400 \, \text{N}}{1100 \, \text{N/m}} = 4 \, \text{m} \]

Step 2: Compute the elastic potential energy stored in the spring

The elastic potential energy (EPE) stored in the spring is given by: \[ \text{EPE} = \frac{1}{2} k x^2 = \frac{1}{2} \times 1100 \, \text{N/m} \times (4 \, \text{m})^2 = 8800 \, \text{J} \]

Step 3: Calculate the work done by friction

The work done against friction is given by the force of friction multiplied by the distance: \[ W_{friction} = f \cdot d = 40 \, \text{N} \times 4 \, \text{m} = 160 \, \text{J} \]

Step 4: Determine gravitational potential energy change

The gravitational potential energy change (GPE) as the performer moves 2.5 m higher is: \[ \Delta \text{GPE} = mgh = 60 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 2.5 \, \text{m} = 1471.5 \, \text{J} \]

Step 5: Use conservation of energy to find final kinetic energy

The initial elastic potential energy is converted into kinetic energy (KE), with some energy lost to friction and increased potential energy. Set up the equation:\[ \text{EPE} - W_{friction} - \Delta \text{GPE} = \text{KE} \]\[ 8800 \, \text{J} - 160 \, \text{J} - 1471.5 \, \text{J} = \text{KE} \]\[ \text{KE} = 7168.5 \, \text{J} \]

Step 6: Solve for the final velocity

The kinetic energy is related to speed by the formula:\[ \text{KE} = \frac{1}{2} mv^2 \]Solving for \( v \):\[ v = \sqrt{\frac{2 \times 7168.5 \, \text{J}}{60 \, \text{kg}}} \approx 15.5 \, \text{m/s} \]

Key Concepts

Energy Conservation

Energy conservation is a vital principle in physics which states that energy cannot be created or destroyed, only transferred or converted from one form to another. In the context of our exercise, the initial elastic potential energy stored in the compressed spring is transferred into The Great Sandini’s kinetic energy as he moves through the barrel.
However, some energy is also transferred to overcome friction and to increase his gravitational potential energy as he rises by 2.5 meters.
The conservation equation used is:

• Initial Elastic Potential Energy = Final Kinetic Energy + Work done against friction + Change in Gravitational Potential Energy

Understanding this equation allows us to determine the speed at which The Great Sandini emerges from the cannon by accounting for all forms of energy transformation involved in his motion.

Kinetic Energy

Kinetic energy is the energy of motion. It's defined by the formula:

• \( \text{KE} = \frac{1}{2} mv^2 \)

where \( m \) is the mass and \( v \) is the velocity. In this exercise, once The Great Sandini exits the spring gun, his kinetic energy is the leftover energy after losses to friction and potential energy changes.
To compute his final speed, we rearrange the formula for kinetic energy to solve for velocity, indicating how fast he travels once all energy transfers are accounted for.

Potential Energy

Potential energy is stored energy that an object has due to its position or state. The primary forms considered here are elastic potential energy, from the compressed spring, and gravitational potential energy, due to The Great Sandini's elevated position.
Gravitational potential energy is given by:

• \( \Delta \text{GPE} = mgh \)

where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height difference. As he rises to a height of 2.5 m, some of the initial energy converts into gravitational potential, reducing the kinetic energy he has available.

Elastic Potential Energy

Elastic potential energy is energy stored in objects that can be stretched or compressed, like springs. The energy stored in the spring is determined by the formula:

• \( ext{EPE} = \frac{1}{2} k x^2 \)

where \( k \) is the spring constant, and \( x \) is the compression distance of the spring.
In the problem, initial compression of the spring captures energy ready to be converted into kinetic motion.
This form of energy is crucial in powering The Great Sandini's launch and must overcome all energy losses to define his speed.

Work Against Friction

Work against friction is the energy required to overcome resistive forces that oppose motion. In this context, it refers to the force needed to move The Great Sandini against the friction within the cannon barrel coated with Teflon.
This work is calculated by:

• \( W_{friction} = f \cdot d \)

where \( f \) is the frictional force and \( d \) is the distance moved. Understanding how friction reduces the total energy available for The Great Sandini's motion helps us appreciate its role in real-world energy conversions.
In our exercise, work done against friction decreases the kinetic energy that could otherwise be used for velocity, thus impacting his speed.