A bow with a spring constant of 398 N/m has an arrow placed in it and pulled back a distance of 0.6 m. If the arrow is shot straight up in the air how high will it go? Assume the mass of the arrow is 0.16 Kg.
To determine how high the arrow will go, we need to use the concept of potential energy and kinetic energy.
First, let's calculate the potential energy stored in the bow when it is pulled back. The potential energy of a spring is given by the formula:
Potential energy = (1/2) * spring constant * displacement^2
where the displacement is the distance the bow is pulled back.
Substituting the given values:
Potential energy = (1/2) * 398 N/m * (0.6 m)^2
Next, let's calculate the potential energy converted into kinetic energy when the arrow is shot. The potential energy is converted into kinetic energy as the arrow leaves the bow. The total energy remains constant.
Since the arrow is shot straight up, when it reaches its highest point, all its potential energy is converted into kinetic energy. The kinetic energy is given by the formula:
Kinetic energy = (1/2) * mass * velocity^2
At the highest point, the kinetic energy is zero because the arrow momentarily stops before falling back down. So, we can equate the initial potential energy to zero kinetic energy:
Potential energy = Kinetic energy
Substituting the known values of potential energy and mass:
(1/2) * 398 N/m * (0.6 m)^2 = (1/2) * 0.16 kg * velocity^2
Now, we can solve for the velocity of the arrow:
velocity^2 = (398 N/m * (0.6 m)^2) / 0.16 kg
Simplifying the equation:
velocity^2 = 897 N / 0.16 kg
velocity^2 = 5606.25 m^2/s^2
Taking the square root of both sides:
velocity = √5606.25 m^2/s^2
velocity ≈ 74.84 m/s
Now that we have the velocity of the arrow, we can calculate how high it will go using the following equation:
Height = velocity^2 / (2 * acceleration due to gravity)
Since the arrow is shot straight up, acceleration due to gravity acts in the opposite direction. Thus, acceleration due to gravity is -9.8 m/s^2.
Substituting the known values:
Height = (74.84 m/s)^2 / (2 * -9.8 m/s^2)
Height ≈ 286.26 m
Therefore, the arrow will reach a height of approximately 286.26 meters when shot straight up into the air.