If the spring is compressed a distance of 0.144 m and the gun fired vertically as shown, the gun can launch a 18.5g projectile from rest to a maximum height of 19.5 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)
well, the Potential energy in the spring must equal the change in PE of the projectile.
1/2 k x^2=mgh
solve for k
Then speed of the projectile at equilibrum
1/2 kx^2=1/2 m v^2 solve for v.
To determine the spring constant, we can use the potential energy stored in the spring formula.
The potential energy stored in a spring is given by:
PE = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
Given that the spring is compressed by a distance of 0.144 m, and the maximum height reached by the projectile is 19.5 m, we can set up the equation as follows:
PE (at maximum height) = PE (compressed spring)
(1/2) * k * (0.144)^2 = (1/2) * k * (0)^2
Since the potential energy at the maximum height is zero, the equation simplifies to:
k * (0.144)^2 = 0
This means that the spring constant k must be zero in order for the equation to be satisfied.
Therefore, the spring constant in this scenario is zero.
Moving on to determining 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.
The total mechanical energy of the projectile at the maximum height is equal to the sum of its kinetic and potential energies. At this point, the potential energy is at its maximum and the kinetic energy is zero.
Therefore, we can use the formula for potential energy at the maximum height:
PE = m * g * h
where m is the mass of the projectile, g is the acceleration due to gravity, and h is the maximum height.
Given that the mass of the projectile is 18.5 g (0.0185 kg) and the maximum height is 19.5 m, we can plug in the values and solve for potential energy:
PE = 0.0185 kg * 9.8 m/s^2 * 19.5 m
= 3.52365 J
Since mechanical energy is conserved, this potential energy is also equal to the initial kinetic energy of the projectile at the equilibrium position of the spring.
The formula for kinetic energy is:
KE = (1/2) * m * v^2
where v is the velocity of the projectile.
Setting the potential energy equal to the kinetic energy, we can solve for v:
3.52365 J = (1/2) * 0.0185 kg * v^2
Rearranging the equation, we get:
v^2 = (2 * 3.52365 J) / 0.0185 kg
= 380.28216 m^2/s^2
Taking the square root of both sides, we find:
v = √(380.28216 m^2/s^2)
≈ 19.5 m/s
Therefore, the speed of the projectile as it moves through the equilibrium position of the spring is approximately 19.5 m/s.
To determine the spring constant, we need to use the conservation of energy principle. The potential energy stored in the compressed spring is converted into the gravitational potential energy gained by the projectile.
The potential energy stored in the spring is given by the formula:
Potential energy = (1/2) * k * x^2
Where k is the spring constant and x is the distance the spring is compressed.
The gravitational potential energy gained by the projectile when it reaches its maximum height can be calculated using the formula:
Gravitational potential energy = m * g * h
Where m is the mass of the projectile, g is the acceleration due to gravity, and h is the maximum height.
Since the potential energy stored in the spring is equal to the gravitational potential energy gained by the projectile, we can set the two equations equal to each other:
(1/2) * k * x^2 = m * g * h
Substituting the given values:
(1/2) * k * (0.144 m)^2 = (0.0185 kg) * (9.8 m/s^2) * (19.5 m)
Simplifying the equation:
k * (0.020736 m^2) = 0.0341535 kg*m^2/s^2
Dividing through by (0.020736 m^2):
k = (0.0341535 kg*m^2/s^2) / (0.020736 m^2)
The calculation results in the spring constant k:
k ≈ 1.647 N/m
To determine the speed of the projectile as it moves through the equilibrium position of the spring, we can use the conservation of mechanical energy. At the equilibrium position, the kinetic energy of the projectile is equal to the potential energy stored in the spring.
The potential energy stored in the spring at equilibrium is given by:
Potential energy = (1/2) * k * x^2
Since the projectile is at rest initially, its kinetic energy is zero. Therefore, we can set the two quantities equal to each other:
(1/2) * k * x^2 = 0
Solving for x, we find:
x = 0
This means that at the equilibrium position, the displacement of the spring is zero. Therefore, the speed of the projectile at this point is also zero.