Question

A spring gun of spring constant \(90 \times 10^{2} \mathrm{~N} / \mathrm{M}\) is compressed \(4 \mathrm{~cm}\) by a ball of mass \(16 \mathrm{~g}\). If the trigger is pulled, calculate the velocity of the ball. (A) \(60 \mathrm{~m} / \mathrm{s}\) (B) \(3 \mathrm{~m} / \mathrm{s}\) (C) \(90 \mathrm{~m} / \mathrm{s}\) (D) \(30 \mathrm{~m} / \mathrm{s}\)

Short answer

The correct answer is (D) \(30 \mathrm{~m} / \mathrm{s}\).

Step-by-step solution

Identify the given values and the unknown

In this problem, we are given: - Spring constant (k) = 90 × 10² N/m - Compression distance (x) = 4 cm (convert to meters) - Mass of the ball (m) = 16 g (convert to kg) - Initial velocity of the ball (vi) = 0 (since the ball is at rest) The unknown we need to find is the final velocity of the ball (vf) after the trigger is pulled.

Convert the given values to their appropriate SI units

First, we need to convert the compression distance (x) and the mass of the ball (m) to their appropriate SI units (meters and kilograms, respectively): - x = 4 cm = 0.04 m - m = 16 g = 0.016 kg

Find the potential energy stored in the compressed spring

Now, let's find the potential energy (PE) stored in the compressed spring using the formula: PE = 1/2 × k × x² PE = 1/2 × (90 × 10² N/m) × (0.04 m)² PE = 72 J (Joules)

Apply the conservation of energy principle

As per the conservation of energy principle, the potential energy stored in the compressed spring will convert to the kinetic energy (KE) of the ball when it is released. Thus, we can write: PE = KE Since the formula for kinetic energy is KE = 1/2 × m × v², we can equate the potential and kinetic energies: 72 J = 1/2 × 0.016 kg × vf²

Solve for the final velocity of the ball (vf)

Now, let's solve for the final velocity of the ball (vf): 72 J = 1/2 × 0.016 kg × vf² vf² = (72 × 2) / 0.016 vf² = 9000 vf = \( \sqrt{9000} \) vf ≈ 30 m/s The velocity of the ball after the trigger is pulled is approximately 30 m/s. Therefore, the correct answer is (D) \(30 \mathrm{~m} / \mathrm{s}\).

Key concepts

Spring Constant

A spring constant, often denoted by the letter \( k \), is a measure of a spring's stiffness. It represents the force required to compress or extend the spring by a unit of length. In this problem, the spring constant is given as \( 90 \times 10^{2} \; \text{N/m} \). This means that for each meter the spring is compressed, it would require a force of \( 9000 \; \text{N} \). The larger the spring constant, the stiffer the spring, making it harder to compress or stretch.

In practical terms, understanding the spring constant is crucial because it determines how much potential energy a spring can store when compressed or extended. This energy is then available to be converted into other forms, such as kinetic energy, when the spring is released.

Potential Energy

Potential energy stored in a spring is the energy held due to its position or configuration. When a spring is compressed or stretched from its rest position, it stores energy. This stored energy is given by the formula:

• \( \text{PE} = \frac{1}{2} k x^2 \)

where \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position (in meters).

In the given exercise, the spring is compressed by \( 4 \; \text{cm} \) (which converts to \( 0.04 \; \text{m} \) for calculations). When we plug the values into the potential energy formula, we find that the spring stores \( 72 \; \text{J} \) of potential energy.

This energy will be converted into kinetic energy when the spring is released, setting the ball into motion. Understanding potential energy helps us see how stored experiences of tension or compression in a spring can do work when released.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed in an isolated system. It can only be transformed from one form to another. In this exercise, the energy transformation happens from potential energy to kinetic energy.
The potential energy stored in the compressed spring is entirely converted into the kinetic energy of the ball once the trigger is released. Thus, we equate the potential energy \( \text{PE} \) with kinetic energy \( \text{KE} \):

• \( \text{PE} = \text{KE} \)

This equation underpins the idea that the total energy before and after the release remains constant under ideal conditions (no friction or air resistance). By applying this principle, we can calculate the velocity of the ball, as all the stored potential energy in the spring is transferred into the motion of the ball.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is determined by the mass and velocity of the object. The formula for kinetic energy is:

• \( \text{KE} = \frac{1}{2} m v^2 \)

where \( m \) is the mass, and \( v \) is the velocity of the object.

For the ball in the spring gun problem, once it is released from the spring, all the potential energy stored is converted into kinetic energy, propelling the ball forward. We computed this using the given mass of the ball (\( 0.016 \; \text{kg} \)) and solved for its velocity through the equation derived from energy conservation. This energy transformation explains how the ball reaches a speed of approximately \( 30 \; \text{m/s} \).

Understanding kinetic energy is essential to grasp how objects in motion utilize energy and how speed and mass play critical roles in determining the energy of moving objects.