A spring gun is made by compressing a spring in a tube and then latching the spring at the compressed position. A 4.97-g pellet is placed against the compressed and latched spring. The spring latches at a compression of 3.92 cm and it takes a force of 9.12 N to compress the spring to that point.
If the gun is fired vertically, how fast (m/s) is the pellet moving when it loses contact with the spring?
The spring constant is
k = 9.12/0.0392 = 232.7 newtons/meter
The potential energy stored in the compressed spring, when compressed a distance X is
(1/2)kX^2 = 0.1788 newtons
When fired vertically, the spring stops exerting a force on the spring when the compression force equals M g. (After that, contact is lost). Call the that position X'
kX' = M g ; so X' = 0.0021 m
At that point,
Pellet kinetic energy + pellet gravitational energy change = spring potentialo energy change
(1/2) M V^2 + Mg(X - X') = (1/2) k [X^2 - X'^2]
Solve for V; check my thinking
The first minus sign on the left should be a plus.
To determine the speed at which the pellet will be moving when it loses contact with the spring, we can use the principle of conservation of mechanical energy.
1. First, let's calculate the potential energy stored in the compressed spring:
- The force required to compress the spring is given as 9.12 N.
- The compression of the spring is given as 3.92 cm, which is equal to 0.0392 m.
- The potential energy stored in the spring at the compressed position is given by the formula:
Potential energy = (1/2) * k * x^2
Where k is the spring constant and x is the compression of the spring.
2. We can calculate the spring constant by rearranging the formula for the force required to compress the spring:
F = k * x
k = F / x
3. Substitute the given values into the formula to find the spring constant:
k = 9.12 N / 0.0392 m
4. Once we have the spring constant, we can calculate the potential energy stored in the spring:
Potential energy = (1/2) * k * x^2
5. Next, let's calculate the kinetic energy of the pellet when it loses contact with the spring:
- The pellet has lost contact with the spring, so all of the stored potential energy is converted into kinetic energy.
- The kinetic energy of the pellet is given by the formula:
Kinetic energy = (1/2) * m * v^2
Where m is the mass of the pellet and v is its velocity.
6. Set the potential energy equal to the kinetic energy to find the velocity of the pellet:
Potential energy = Kinetic energy
(1/2) * k * x^2 = (1/2) * m * v^2
7. Rearrange the equation to solve for velocity:
v = sqrt((k * x^2) / m)
8. Substitute the given values into the equation to find the velocity of the pellet:
v = sqrt((k * x^2) / m)
Now, we can plug in the given values and calculate the velocity of the pellet:
Mass of the pellet (m) = 4.97 g = 0.00497 kg
Compression of the spring (x) = 3.92 cm = 0.0392 m
Force required to compress the spring (F) = 9.12 N
Using the derived spring constant (k), we can substitute the values and calculate the velocity of the pellet.
To find the speed at which the pellet loses contact with the spring, we can use the principle of conservation of mechanical energy. The potential energy stored in the compressed spring will be converted into kinetic energy as the pellet is launched.
First, let's find the potential energy stored in the compressed spring. The potential energy (PE) of a spring can be calculated using the formula:
PE = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring.
To find the spring constant (k), we can use Hooke's law, which states that the force required to compress or extend a spring is directly proportional to the displacement:
F = k * x
Rearranging the equation, we get:
k = F / x
where F is the force applied and x is the displacement.
Given that the force applied to compress the spring is 9.12 N and the displacement is 3.92 cm (or 0.0392 m), we can calculate the spring constant:
k = 9.12 N / 0.0392 m = 232.65 N/m
Now we can substitute the spring constant and displacement into the potential energy equation:
PE = (1/2) * (232.65 N/m) * (0.0392 m)^2
PE = 0.044628 J
Now we can equate the potential energy stored in the spring to the kinetic energy of the pellet when it loses contact with the spring. The kinetic energy (KE) can be calculated using the equation:
KE = (1/2) * m * v^2
where m is the mass of the pellet and v is its velocity.
Given that the mass of the pellet is 4.97 g (or 0.00497 kg), we can equate the potential and kinetic energy equations:
0.044628 J = (1/2) * (0.00497 kg) * v^2
Now we can solve for v:
v^2 = (2 * 0.044628 J) / (0.00497 kg)
v^2 = 89.903 m^2/s^2
v = √(89.903 m^2/s^2)
v = 9.49 m/s (rounded to two decimal places)
Therefore, the pellet will be moving at approximately 9.49 m/s when it loses contact with the spring.