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 5 cm, and it takes a force of
9.12 N to compress the spring to that point.
(a) If the gun is �red vertically, how fast (m/s) is the pellet moving when it loses contact
with the spring?
(b) To what maximum height (m) will the pellet rise? (as measured from the original
latched position)
The spring constant is
k = 9.12/0.05 = 182.5 N/m
Contact ceases when the compression is x = 0, when no more force is applied by the spring. At that time,
(1/2)kXo^2 = (1/2) m V^2
Xo is the latched compression, 0.05 m.
Solve for V. m must be in kilograms.
2) From the original latched position, add Xo to sqrt(Vo^2)/(2g)
To answer the given questions, we need to apply the concepts of work, potential energy, and kinetic energy.
(a) To find the velocity of the pellet when it loses contact with the spring, we can equate the potential energy stored in the compressed spring to the kinetic energy of the pellet.
1. Determine the potential energy stored in the compressed spring:
The potential energy (PE) stored in a spring can be calculated using the formula:
PE = (1/2)kx^2,
where k is the spring constant and x is the compression of the spring.
Given:
Force applied (F) = 9.12 N
Compression of the spring (x) = 5 cm = 0.05 m
Since the force required to compress the spring can be expressed as:
F = kx,
we can solve for k as:
k = F/x
k = 9.12 N / 0.05 m = 182.4 N/m
Using this value, we can calculate the potential energy in the spring:
PE = (1/2)(182.4 N/m)(0.05 m)^2 = 0.455 J
2. Equate the potential energy in the spring to the kinetic energy of the pellet:
At the moment the pellet loses contact with the spring, all of the potential energy is converted into kinetic energy. The kinetic energy (KE) of an object can be calculated using the formula:
KE = (1/2)mv^2,
where m is the mass of the object and v is its velocity.
Given:
Mass of the pellet (m) = 4.97 g = 0.00497 kg
Setting the potential energy equal to the kinetic energy:
KE = PE
(1/2)mv^2 = 0.455 J
Solving for v:
v^2 = (2)(0.455 J) / 0.00497 kg
v^2 = 183.1 m^2/s^2
v ≈ 13.5 m/s
Therefore, the pellet is moving with a velocity of approximately 13.5 m/s when it loses contact with the spring.
(b) To determine the maximum height the pellet will reach, we can use the principle of conservation of mechanical energy.
1. Calculate the maximum height using the conservation of mechanical energy:
At the highest point, all the initial kinetic energy will be converted into potential energy. Therefore, the initial kinetic energy (0.455 J) is equal to the potential energy at the maximum height.
Given:
g = acceleration due to gravity = 9.8 m/s^2
Using the formula for potential energy:
PE = mgh,
where m is the mass, g is the acceleration due to gravity, and h is the height.
Setting the potential energy equal to the initial kinetic energy:
0.455 J = (0.00497 kg)(9.8 m/s^2)h
h = 0.455 J / (0.00497 kg)(9.8 m/s^2)
h ≈ 9.19 m
Therefore, the pellet will rise to a maximum height of approximately 9.19 meters from the original latched position.