If the spring is compressed a distance of 0.150 m and the gun fired vertically as shown, the gun can launch a 27.6g projectile from rest to a maximum height of 16.1 m above the starting point of the projectile. Neglecting all resistive forces, determine the spring constant.
Determine the speed of the projectile as it moves through the equilibrium position of the spring (where x = 0) (as shown in (b)).
You know the energy in the spring:
PE=massprojetile*g*heightitgoes
so now in the spring.
PE=1/2 k x^2
solve for k
speed through equilibrium? 1/2 mv^2=PE above solve for v
To determine the speed of the projectile as it moves through the equilibrium position of the spring, we need to consider the conservation of mechanical energy.
First, let's determine the potential energy of the projectile at the maximum height. Since the projectile is at rest at this point, its kinetic energy is zero. Therefore, all of its initial energy is in the form of potential energy.
Potential energy (PE) can be calculated using the formula PE = mgh, where m is the mass of the projectile (27.6 g or 0.0276 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (16.1 m).
PE = (0.0276 kg)(9.8 m/s^2)(16.1 m)
PE = 4.465 Joules
Now, let's determine the potential energy of the compressed spring. The potential energy of a spring can be calculated using the formula PE = (1/2)kx^2, where k is the spring constant and x is the compression distance (0.150 m).
PE = (1/2)k(0.150 m)^2
PE = 0.01125k Joules
Since mechanical energy is conserved, we can equate the potential energy at the maximum height (4.465 J) to the potential energy of the compressed spring (0.01125k J):
4.465 J = 0.01125k J
Now, we can solve for the spring constant:
k = (4.465 J)/(0.01125 J)
k ≈ 396.44 N/m
Therefore, the spring constant is approximately 396.44 N/m.
To determine the speed of the projectile as it moves through the equilibrium position of the spring (where x = 0), we can use the principle of conservation of mechanical energy.
At the equilibrium position, all of the potential energy stored in the compressed spring is converted into kinetic energy of the projectile.
The potential energy of the compressed spring can be calculated using the formula PE = (1/2)kx^2, where k is the spring constant and x is the compression distance (0 m at the equilibrium position).
PE = (1/2)(396.44 N/m)(0 m)^2
PE = 0 J
Since all the potential energy is converted into kinetic energy at this point, the kinetic energy (KE) of the projectile at the equilibrium position is equal to the potential energy of the compressed spring:
KE = PE = 0 J
The kinetic energy of a moving object can be calculated using the formula KE = (1/2)mv^2, where m is the mass of the projectile (0.0276 kg) and v is its velocity.
0 J = (1/2)(0.0276 kg)v^2
Solving for v:
0 = (1/2)(0.0276 kg)v^2
As the equation is quadratic, we solve for v^2:
0 = (1/2)(0.0276 kg)v^2
0 = (1/2)(0.0276 kg)v^2
0 = (0.0138 kg)v^2
To find the speed v, we need to take the square root of both sides:
0 = 0.0138v^2
Therefore, the speed of the projectile as it moves through the equilibrium position of the spring is 0 m/s.