A mass of 2.2kg is placed on a stiff vertical spring which has a spring constant of 950n/m. The object is then pressed against the spring until it has been compressed a distance of 77cm. The mass is then released and is allowed to be thrown up into the air.
How high in the air will the mass be thrown? (13.05m)
What will be the velocity of the mass just as it leaves the end of the spring? (15.5m/s)
13.05 meters
To find the height to which the mass will be thrown and the velocity at which it leaves the spring, we can break down the problem into two parts: the potential energy stored in the compressed spring and the conversion of that potential energy into kinetic energy.
1. Calculate the potential energy stored in the compressed spring:
The potential energy stored in a spring is given by the equation:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
In this case, the spring constant (k) is given as 950 N/m, and the displacement (x) is 0.77 m (convert 77 cm to meters). Plugging these values into the equation, we get:
Potential Energy = (1/2) * 950 N/m * (0.77 m)^2.
Calculating this expression, we find the potential energy stored in the compressed spring is 287.537 J (Joules).
2. Convert the potential energy into kinetic energy:
At the moment the mass leaves the spring, all of the potential energy stored in the spring gets converted into kinetic energy.
Kinetic Energy = Potential Energy
3. Calculate the height to which the mass is thrown:
The potential energy converted into kinetic energy is equal to the kinetic energy gained by the mass as it is thrown upwards. The kinetic energy of an object is given by the equation:
Kinetic Energy = (1/2) * m * v^2
where m is the mass and v is the velocity.
In this case, the mass (m) is given as 2.2 kg. We need to solve for v, the velocity of the mass just as it leaves the end of the spring. Set the potential energy (287.537 J) equal to the kinetic energy expression:
287.537 J = (1/2) * 2.2 kg * v^2.
Rearranging the equation to solve for v, we get:
v = sqrt(2 * 287.537 J / 2.2 kg).
Evaluating this expression, we find the velocity of the mass just as it leaves the end of the spring is approximately 15.5 m/s (rounded to one decimal place).
4. Calculate the height to which the mass is thrown:
To find the height to which the mass is thrown, we can use the equation for vertical projectile motion:
Final Height = (Initial Velocity^2) / (2 * Gravity)
where gravity is the acceleration due to gravity (approximated as 9.8 m/s^2).
Using the velocity we found (15.5 m/s) and the acceleration due to gravity (9.8 m/s^2), we can calculate the height:
Final Height = (15.5 m/s)^2 / (2 * 9.8 m/s^2).
Evaluating this expression, we find that the mass will be thrown to a height of approximately 15.8 m (rounded to two decimal places).
Thus, the mass will be thrown to a height of approximately 15.8 meters and the velocity at which it leaves the end of the spring is approximately 15.5 m/s.