You shoot a 56 gram dart vertically upwards from a catapult with a spring constant of 335 N/m. The catapult is initially stretched from the equilibrium point by 27 cm. What is the height above the starting point reached by the dart?
13.1
To find the height reached by the dart, you can use the principles of energy conservation. The potential energy stored in the spring of the catapult is converted into the potential energy of the dart at its highest point.
1. Find the potential energy stored in the spring:
The potential energy stored in a spring is given by the formula:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
In this case, the spring constant is 335 N/m and the displacement is 0.27 m (27 cm converted to meters). Plugging these values into the formula:
Potential Energy = (1/2) * 335 N/m * (0.27 m)^2
2. Find the potential energy of the dart at its highest point:
At the highest point, all of the potential energy of the spring is converted into the potential energy of the dart. The potential energy of an object at height h is given by its mass multiplied by the acceleration due to gravity (g) multiplied by its height.
Potential Energy = m * g * h
In this case, the mass of the dart is 56 grams, which is equivalent to 0.056 kg. The acceleration due to gravity is approximately 9.8 m/s^2. We want to solve for h, which is the height reached by the dart.
Set the potential energy of the dart equal to the potential energy stored in the spring and solve for h:
(1/2) * 335 N/m * (0.27 m)^2 = 0.056 kg * 9.8 m/s^2 * h
Simplifying and solving for h:
h = [(1/2) * 335 N/m * (0.27 m)^2] / [0.056 kg * 9.8 m/s^2]
Calculate the value of h to find the height reached by the dart.