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Chapter 4: Problem 431

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

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

Step by step solution

01

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.
02

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
03

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)
04

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
05

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) \(30 \mathrm{~m} / \mathrm{s}\).

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