A 439g block is placed on a spring with a spring constant of 193N/m, compressing it 28cm. What height does the block reach when it is launched?

To determine the height the block reaches when it is launched, we need to consider the conservation of energy.

First, let's find the potential energy stored in the compressed spring. The equation for the potential energy of a spring is given by:

Potential Energy (PE) = (1/2) * k * x^2

where:
k is the spring constant (193 N/m)
x is the compression of the spring (28 cm = 0.28 m)

Substituting the values into the equation:

PE = (1/2) * 193 * (0.28)^2
PE = 9.726 J

This potential energy will be converted into kinetic energy when the block is launched. The equation for kinetic energy is given by:

Kinetic Energy (KE) = (1/2) * mass * velocity^2

At the highest point of the block's trajectory, it will briefly come to rest before falling back down. Therefore, all of the potential energy will be converted to gravitational potential energy at this point. The equation for gravitational potential energy is given by:

Gravitational Potential Energy (GPE) = mass * g * h

where:
mass is the mass of the block (439 g = 0.439 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height reached by the block

Setting the potential energy equal to the gravitational potential energy:

PE = GPE
9.726 J = 0.439 kg * 9.8 m/s^2 * h
h = 9.726 J / (0.439 kg * 9.8 m/s^2)
h ≈ 2.166 m

Therefore, the block will reach a height of approximately 2.166 meters when it is launched.