Question

The Great Sandini is a 60-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 N/m that he will compress with a force of 4400 N. The inside of the gun barrel is coated with Teflon, so the average friction force will be only 40 N during the 4.0 m he moves in the barrel. At what speed will he emerge from the end of the barrel, 2.5 m above his initial rest position?

Short answer

The Great Sandini will emerge at approximately 15.5 m/s.

Step-by-step solution

Determine Spring Compression

First, use Hooke's law to find the compression of the spring \( x \). Hooke's law is given by \( F = kx \), where \( F \) is the force applied (4400 N) and \( k \) is the spring constant (1100 N/m). Solving for \( x \): \[x = \frac{F}{k} = \frac{4400\, \text{N}}{1100\, \text{N/m}} = 4\, \text{m}.\]

Calculate Spring Potential Energy

The potential energy stored in the spring (\( U_{s} \)) when compressed is given by \( U_{s} = \frac{1}{2} k x^2 \). Substitute \( k = 1100 \text{ N/m} \) and \( x = 4 \text{ m} \): \[U_{s} = \frac{1}{2} \times 1100 \times (4)^2 = 8800\, \text{J}.\]

Determine Work Done by Friction

The work done by the friction force (\( W_f \)) can be calculated using \( W_f = f \times d \), where \( f \) is the friction force (40 N) and \( d \) is the distance over which the force acts (4.0 m). Therefore: \[W_f = 40\, \text{N} \times 4.0\, \text{m} = 160\, \text{J}.\]

Calculate Gravitational Potential Energy Change

The change in gravitational potential energy (\( \Delta U_g \)) as The Great Sandini moves up 2.5 meters is given by \( \Delta U_g = mgh \), where \( m = 60\, \text{kg} \), \( g = 9.8\, \text{m/s}^2 \), and \( h = 2.5\, \text{m} \). Thus: \[\Delta U_g = 60 \times 9.8 \times 2.5 = 1470\, \text{J}.\]

Apply Energy Conservation

The energy conservation equation for this problem is: Initial Spring Energy = Kinetic Energy + Friction Work + Gravitational Potential Energy Change. Let the final kinetic energy be \( K_f \):\[ 8800 = K_f + 160 + 1470.\] Thus, solving for \( K_f \):\[K_f = 8800 - 160 - 1470 = 7170\, \text{J}.\]

Solve for Final Speed

The kinetic energy of The Great Sandini at the end of the barrel when he emerges is given by \( K_f = \frac{1}{2} mv^2 \). Substituting \( K_f = 7170\, \text{J} \) and \( m = 60\, \text{kg} \), and solving for \( v \): \[7170 = \frac{1}{2} \times 60 \times v^2.\]Rearranging for \( v \): \[v^2 = \frac{2 \times 7170}{60} = 239,\]\[v = \sqrt{239} \approx 15.5\, \text{m/s}.\]

Final Step: Conclude with the Speed

The Great Sandini will emerge with a speed of approximately 15.5 m/s after taking into account the spring compression, friction, and elevation change.

Key concepts

Spring Potential Energy

When a spring is compressed or stretched, it stores energy known as spring potential energy. This is a crucial concept in physics as it demonstrates how potential energy gets converted into kinetic energy, propelling an object. The potential energy stored in a spring can be calculated using the formula:

• \( U_s = \frac{1}{2} k x^2 \)

Here, \( k \) is the spring constant, which defines the spring's stiffness, and \( x \) is the displacement, which is how much the spring is compressed or stretched.
In our exercise, the spring constant is 1100 N/m, and it’s compressed by 4 m, resulting in a spring potential energy of 8800 J. This amount of energy will initially propel The Great Sandini forward, acting as the primary source of his kinetic energy. Understanding this exchange of energy is essential in physics problem solving.

Energy Conservation

The law of energy conservation plays a central role in physics, asserting that energy cannot be created or destroyed, only transformed from one form to another. In our particular scenario, the energy begins as spring potential energy and is ultimately converted into kinetic energy, with some losses due to friction and the increase in gravitational potential energy.
This principle can be illustrated using the equation:

• Initial Energy = Final Energy + Energy Losses

In this exercise, the initial spring energy is set to equal the sum of kinetic energy when The Great Sandini exits the barrel, the work done by friction, and the change in gravitational potential energy. By applying this principle, we ensure that all forms of energy throughout the process are accounted for, providing a comprehensive understanding of the system's dynamics.

Friction Work

Friction is a force that opposes motion, and in this context, it's an energy loss that takes away from the potential energy provided by the spring. The work done by friction can be calculated using the formula:

• \( W_f = f \times d \)

where \( f \) is the friction force, and \( d \) is the distance over which this force is applied.
In the case of The Great Sandini, the friction force is 40 N, and it acts over a 4 m distance, resulting in a friction work of 160 J. This energy, subtracted from the initial spring energy, represents the portion of energy lost due to the opposing force of the Teflon-coated barrel.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its height above the ground. This energy is calculated through the formula:

• \( \Delta U_g = mgh \)

Here, \( m \) is mass, \( g \) is the acceleration due to gravity (approximately 9.8 m/s²), and \( h \) is the height above the starting point.
For our circus performer, moving up 2.5 meters while weighing 60 kg, the change in gravitational potential energy is 1470 J. This increase in potential energy while ascending reduces the kinetic energy available for maintaining speed, further emphasizing the interplay between different types of energy in closed systems.

Kinetic Energy Calculation

Kinetic energy represents the energy of motion. After the conversion of spring potential energy and accounting for energy losses due to friction and gravity, The Great Sandini's speed can be calculated using kinetic energy. The formula is:

• \( K_f = \frac{1}{2} mv^2 \)

where \( m \) is mass and \( v \) is velocity.
In this exercise, once you defeat frictional and gravitational challenges, you find that the remaining kinetic energy is 7170 J. Substituting this value, along with Sandini’s mass of 60 kg, into the kinetic energy formula allows us to solve for \( v \), yielding a velocity of approximately 15.5 m/s. This final step demonstrates how potential energy transformations and energy losses interact to define movement and speed.