A toy "spring gun" has a spring with a spring constant of 30,000 N/m and is depressed 4 cm. It shoots a plastic dart with a mass of 45 grams. How fast will the dart travel if there is no friction?
KE = PE
1/2 mv^2 = 1/2 kx^2
Since k is in N/m, we need to use MKS units.
1/2 * 0.045 * v^2 = 1/2 * 30000 * 0.04^2
v = 32.7 m/s
well, the KE of the dart will equal the PE stored in the spring, right?
I'm not really grasping this concept well. I don't know if the KE of the dart is equal is the PE stored? Based on the formulas I don't know that is correct.
PE= 1/2kx^2
k=30,000
x=4cm
KE= 1/2mv^2
m=45g
v=0??
where else is the stored spring energy going?
the problem says no friction
ooooh for some reason I thought I needed to subtract PE and KE and it wasn't making any sense.
To determine the speed at which the dart will travel, we can use the principle of conservation of mechanical energy. We assume that the potential energy stored in the compressed spring is converted entirely into kinetic energy of the dart.
Step 1: Convert the given mass from grams to kilograms.
- The mass of the dart is 45 grams, which can be converted to 0.045 kilograms (by dividing by 1000).
Step 2: Calculate the potential energy stored in the compressed spring.
- The potential energy stored in a spring is given by the formula: PE = (1/2)kx^2
- where PE is the potential energy, k is the spring constant, and x is the displacement or compression of the spring.
- In this case, the spring constant is 30,000 N/m, and the spring is depressed 4 cm which can be converted to 0.04 meters.
- Substituting the values into the formula: PE = (1/2)(30,000)(0.04^2)
- Evaluating the expression: PE = (1/2)(30,000)(0.0016) = 24 Joules
Step 3: Apply the principle of conservation of mechanical energy.
- The potential energy stored in the spring at its maximum compression is equal to the kinetic energy gained by the dart when it is released.
- So, the kinetic energy of the dart can be calculated as KE = PE = 24 Joules.
Step 4: Calculate the speed of the dart using the kinetic energy.
- The kinetic energy can be calculated using the formula: KE = (1/2)mv^2
- where KE is the kinetic energy, m is the mass of the dart, and v is the velocity of the dart.
- In this case, the kinetic energy is 24 Joules and the mass of the dart is 0.045 kg.
- Substituting the values into the formula: 24 = (1/2)(0.045)(v^2)
- Solving for v, we get: v^2 = (24*2) / 0.045
- Evaluating the expression: v^2 = 1066.67
- Taking the square root of both sides to find the velocity: v = √1066.67
- Evaluating the expression: v ≈ 32.67 m/s
Therefore, if there is no friction, the dart will travel at approximately 32.67 meters per second.