A vertical spring (ignore its mass), whose spring constant is 875 N/m, is attached to a table and is compressed down by 0.220 m. What upward speed can it give to a 0.380-kg ball when released?

Verified step by step guidance

1. Identify the energy stored in the compressed spring using the formula for elastic potential energy: $E\_s\; =\; \backslash (\backslash frac\{1\}\{2\}\backslash )\; k\; x^2$, where $k$ is the spring constant (875 N/m) and $x$ is the compression distance (0.220 m).
2. Recognize that when the spring is released, the stored elastic potential energy is converted into the kinetic energy of the ball. Use the kinetic energy formula: $E\_k\; =\; \backslash (\backslash frac\{1\}\{2\}\backslash )\; m\; v^2$, where $m$ is the mass of the ball (0.380 kg) and $v$ is its velocity.
3. Set the elastic potential energy equal to the kinetic energy to apply the conservation of energy principle: $\backslash (\backslash frac\{1\}\{2\}\backslash )\; k\; x^2\; =\; \backslash (\backslash frac\{1\}\{2\}\backslash )\; m\; v^2$.
4. Simplify the equation to solve for the velocity $v$: $v\; =\; \backslash (\backslash sqrt\{\backslash frac\{k\; x^2\}\{m\}\backslash )\}$.
5. Substitute the given values ($k\; =\; 875\; \backslash \; \backslash (\backslash text\{N/m\}\backslash )$, $x\; =\; 0.220\; \backslash \; \backslash (\backslash text\{m\}\backslash )$, and $m\; =\; 0.380\; \backslash \; \backslash (\backslash text\{kg\}\backslash )$) into the equation to calculate the upward speed of the ball.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. In this scenario, the spring constant is 875 N/m, and the spring is compressed by 0.220 m, allowing us to calculate the force exerted by the spring when released.

Potential Energy in Springs

The potential energy stored in a compressed or stretched spring is given by the formula PE = 1/2 kx², where k is the spring constant and x is the displacement. This potential energy is converted into kinetic energy when the spring is released, which is crucial for determining the speed imparted to the ball.

Kinetic Energy and Motion

Kinetic energy (KE) is the energy of an object in motion, calculated using the formula KE = 1/2 mv², where m is the mass and v is the velocity. When the spring releases its stored potential energy, it transforms into kinetic energy, allowing us to find the upward speed of the ball by equating the potential energy of the spring to the kinetic energy of the ball.