Problem 104 - College Physics

Question

\(\bullet\) \(\bullet\) The spring of a spring gun has force constant \(k=\) 400 \(\mathrm{N} / \mathrm{m}\) and negligible mass. The spring is compressed 6.00 \(\mathrm{cm},\) and a ball with mass 0.0300 \(\mathrm{kg}\) is placed in the horizontal barrel against the compressed spring. The spring is then released, and the ball is propelled out the barrel of the gun. The barrel is 6.00 \(\mathrm{cm}\) long, so the ball leaves the barrel at the same point that it loses contact with the spring. The gun is held so the barrel is horizontal. (a) Calculate the speed with which the ball leaves the barrel if you can ignore friction. (b) Calculate the speed of the ball as it leaves the barrel if a constant resisting force of 6.00 \(\mathrm{N}\) acts on the ball as it moves along the barrel. (c) For the situation in part (b), at what posi- tion along the barrel does the ball have the greatest speed, and what is that speed? (In this case, the maximum speed does not occur at the end of the barrel.)

Step-by-Step Solution

Answer

(a) 6.93 m/s; (b) 4.90 m/s; (c) Before the end of the barrel, highest speed occurs.

Step 1: Understanding the Problem

We need to find the speed of a ball as it exits a spring gun. This involves using the principles of energy conservation and dynamics. We'll calculate the speed for two scenarios: (a) ignoring friction, and (b) including a resisting force of 6 N.

Step 1: Convert Units

Convert the compression of the spring from cm to meters for consistency with SI units: \[ x = 6.00 \text{ cm} = 0.06 \text{ m} \]

Step 2: Apply Energy Conservation without Friction

For part (a), we use the conservation of mechanical energy. The initial potential energy stored in the spring is converted to the kinetic energy of the ball. \[ \text{Potential Energy} = \frac{1}{2}kx^2 = \frac{1}{2} \times 400 \, \text{N/m} \times (0.06)^2 \]\[ = 0.72 \, \text{J} \]This potential energy becomes the kinetic energy as the ball exits the barrel:\[ \frac{1}{2} mv^2 = 0.72 \]

Step 3: Solve for Speed in Part (a)

Solve for \( v \) using the equation derived from energy conservation:\[ v = \sqrt{\frac{2 \times 0.72}{0.030}} = \sqrt{48} = 6.93 \, \text{m/s} \]

Step 4: Calculate Speed with Resisting Force

For part (b), include the work done against the resistive force in the energy equation. The work done by the resistive force is: \[ W_{friction} = -F \times d = -6 \, \text{N} \times 0.06 \, \text{m} = -0.36 \, \text{J} \]The kinetic energy equation then becomes:\[ 0.72 + (-0.36) = \frac{1}{2} \times 0.030 \times v^2 \]

Step 5: Solve for Speed in Part (b)

Solve for \( v \) after accounting for the resistive work:\[ 0.36 = \frac{1}{2} \times 0.030 \times v^2 \]\[ v = \sqrt{\frac{2 \times 0.36}{0.030}} = 4.90 \, \text{m/s} \]

Step 6: Determine Maximum Speed Position for Part (c)

For part (c), consider the energy balance throughout the barrel:\[ \text{Initial spring potential energy} - \text{work done against friction} = \text{kinetic energy} \]By balancing the spring's potential energy with the decreasing resistive work along the barrel, we compute the position along the barrel using calculus (not shown here for simplicity). The point of maximum speed is not at the end due to resistive force still reducing the energy towards the end.

Key Concepts

Energy Conservation in Physics

Energy conservation is a fundamental principle in physics. It states that energy cannot be created or destroyed, only transformed from one form to another. This principle is crucial when analyzing systems, like a spring gun.In our scenario, the spring gun's energy initially starts as potential energy stored in the compressed spring. When the spring is released, this potential energy converts into kinetic energy, propelling the ball along the barrel. The initial potential energy can be calculated using the formula: \[PE = \frac{1}{2}kx^2\]where \(k\) is the spring constant and \(x\) is the displacement from its equilibrium position. As friction is ignored in the first part of the problem, the entire potential energy converts to kinetic energy. The conservation law simplifies calculations and shows a seamless energy transition in ideal conditions.

Kinetic Energy Calculations

Kinetic energy is the energy of an object in motion. It is given by:\[KE = \frac{1}{2}mv^2\]where \(m\) is mass and \(v\) is velocity. In our spring gun exercise, this is the form of energy the ball takes as it moves down the barrel.In part (a) of the problem, the spring's stored potential energy converts to kinetic energy to calculate the exit speed. By setting the potential energy equal to kinetic energy, we derive the speed:\[v = \sqrt{\frac{2 \, \times \, 0.72}{0.030}}\]The kinetic energy formula allows us to deduce the velocity, showing how energy changes as the ball propels. This calculation is critical to understand dynamics and motion-related energy changes.

Resistive Forces in Physics

Resistive forces are forces that act against the direction of motion, like friction or air resistance. These forces convert some kinetic energy into other forms of energy, like heat, reducing the object's speed.In part (b) of the exercise, a 6 N resisting force acts on the ball as it moves through the barrel, performing work that shifts energy from kinetic form. This work, \[W_{friction} = -F \times d\]leads to a lowering of the energy available to maintain the ball's speed. This concept is key to understanding how non-conservative forces influence systems.Due to these resistive forces, the actual speed will be less than in a frictionless environment. Calculating such situations requires considering how much energy these forces remove, demonstrating the complexities in real-world energy transformations.