A spring gun is made by compressing a spring (assumed to be perfect) and latching it. A spring of constant k = 60 N/ m is used and the latch is located at a distance of 7 cm from equilibrium. The pellets have mass 4 g. What is the muzzle velocity of the gun?
Assume that (1/2) M V^2 = (1/2) k X^2
which means that all of the spring energy is converted to kinetic energy of the pellet, which has mass 0.004 kg.
The spring compression is X = 0.07 m
V = sqrt(k/M)*X
The drawing shows two transverse waves traveling on two strings. The linear density of each string is 0.0483 kg/m, and the tension is provided by a 15.0-N block that is hanging from the string. Determine the speed of the wave in part (a) and part (b) of the drawing.
Why did the spring go to therapy? Because it had a lot of "tension" and needed to work things out!
To find the muzzle velocity of the gun, we can use the principle of conservation of mechanical energy. The potential energy stored in the compressed spring will be converted into kinetic energy of the pellet when it is released.
Step 1: Calculate the potential energy stored in the compressed spring.
Potential energy (PE) stored in a spring is given by the formula:
PE = (1/2) * k * x^2
where k is the spring constant and x is the displacement from equilibrium.
Given:
Spring constant k = 60 N/m
Displacement x = 7 cm = 0.07 m
PE = (1/2) * (60 N/m) * (0.07 m)^2
PE = 0.147 J (Joules)
Step 2: Calculate the kinetic energy of the pellet.
The kinetic energy (KE) of the pellet is given by the formula:
KE = (1/2) * m * v^2
where m is the mass of the pellet and v is its velocity.
Given:
Mass m = 4 g = 0.004 kg
Using the conservation of mechanical energy, the potential energy stored in the spring is converted into kinetic energy of the pellet:
PE = KE
0.147 J = (1/2) * (0.004 kg) * v^2
0.147 J = 0.002 kg * v^2
Simplifying, we find:
v^2 = 0.147 J / 0.002 kg
v^2 = 73.5 m^2/s^2
Step 3: Calculate the muzzle velocity.
Taking the square root of both sides:
v = √73.5 m^2/s^2
v ≈ 8.57 m/s
Therefore, the muzzle velocity of the gun is approximately 8.57 m/s.
To find the muzzle velocity of the gun, we need to apply the principle of conservation of mechanical energy. The energy stored in the compressed spring is converted into the kinetic energy of the pellet as it is released from the gun.
First, we need to calculate the potential energy stored in the compressed spring. The equation for potential energy stored in a spring is:
Potential energy (PE) = (1/2) * k * x^2
Where k is the spring constant, and x is the displacement of the spring from its equilibrium position. In this case, k = 60 N/m and x = 0.07 m.
Plugging in the values, we get:
PE = (1/2) * 60 * (0.07)^2 = 0.147 J (Joules)
Now, we know that this potential energy is converted into kinetic energy. The equation for kinetic energy is:
Kinetic energy (KE) = (1/2) * m * v^2
Where m is the mass of the pellet and v is its velocity.
In this case, the mass of the pellet is 4 g, which is equivalent to 0.004 kg.
Using the conservation of mechanical energy, we can equate the potential energy to the kinetic energy:
PE = KE
0.147 J = (1/2) * 0.004 * v^2
Simplifying, we find:
v^2 = (2 * 0.147 J) / 0.004 kg
v^2 = 73.5 J/kg
Taking the square root of both sides, we find:
v ≈ 8.57 m/s
Therefore, the muzzle velocity of the gun is approximately 8.57 m/s.