A 659g block is placed on a spring with a spring constant of 194N/m, compressing it 30cm. What height does the block reach when it is launched?
To find the height that the block reaches when it is launched, we can use the concept of conservation of mechanical energy.
When the block is launched, it will have both potential and kinetic energy. The potential energy is given by the formula PE = mgh, where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height above the reference point where we want to measure the height.
The kinetic energy is given by the formula KE = (1/2)mv^2, where m is the mass of the block and v is the velocity of the block.
In this problem, the block is initially compressed on the spring, so its potential energy is stored in the spring. When the spring is released, this potential energy is converted into kinetic energy and potential energy due to height.
First, let's find the potential energy stored in the compressed spring. The formula for potential energy in a spring is PE = (1/2)kx^2, where k is the spring constant and x is the distance the spring is compressed.
PE = (1/2)kx^2
= (1/2)(194 N/m)(0.3 m)^2
= 8.73 J (Joules)
The potential energy stored in the spring is 8.73 Joules.
Next, we can set up an equation to find the velocity of the block when it leaves the spring. The potential energy stored in the spring is converted into kinetic energy. Therefore, the potential energy stored in the spring is equal to the kinetic energy.
PE = KE
8.73 J = (1/2)mv^2
We can rearrange the equation to solve for v:
v^2 = (2 * PE) / m
v = sqrt((2 * PE) / m)
= sqrt((2 * 8.73 J) / 0.659 kg)
= 6.03 m/s (rounded to two decimal places)
So, the velocity of the block when it leaves the spring is 6.03 m/s.
Now that we have the velocity of the block, we can use it to find the height it reaches using the formula for potential energy:
PE = mgh
Solving for h:
h = PE / (mg)
= (8.73 J) / (0.659 kg * 9.8 m/s^2)
= 1.34 m (rounded to two decimal places)
Therefore, the height that the block reaches when it is launched is 1.34 meters.