Dana has set up a target for her toy spring gun. The gun requires 5.50 g pellets as ammunition that travel 11.4 m/s when shot. If the spring inside has maximum compression of 0.0700 m, what is the spring constant for the spring in the toy gun?

(1/2) k x^2 = (1/2) m v^2

k (.07)^2 = .0055 (11.4)^2

To find the spring constant, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement.

The formula for Hooke's Law is:

F = -k * x

Where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.

In this case, we know the mass of the pellets (5.50 g) and the velocity at which they are shot (11.4 m/s). We can use these values to calculate the force exerted by the pellets.

First, we need to convert the mass of the pellets to kilograms:

mass = 5.50 g = 0.00550 kg

Next, we can calculate the force exerted by the pellets using Newton's second law of motion:

F = m * a

Since there is no acceleration acting on the pellets once they are shot, we can use the formula:

F = m * v

F = 0.00550 kg * 11.4 m/s

F = 0.0627 N

Now, we can use Hooke's Law to find the spring constant.

F = -k * x

Plugging in the known values:

0.0627 N = -k * 0.0700 m

Solving for k:

k = -0.0627 N / 0.0700 m

k ≈ -0.896 N/m

Therefore, the spring constant for the spring in the toy gun is approximately -0.896 N/m. Note that the negative sign indicates that the force exerted by the spring is in the opposite direction of its displacement.

To find the spring constant (k) for the spring in the toy gun, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed.

Hooke's Law is given by the equation:
F = -kx

Where:
F is the force exerted by the spring
k is the spring constant
x is the displacement/compression of the spring

In this case, the spring has a maximum compression of 0.0700 m.

The force exerted by the spring can be calculated using the kinetic energy of the pellets when shot.

The kinetic energy (KE) of an object is given by the equation:
KE = (1/2)mv²

Where:
m is the mass of the pellets
v is the velocity of the pellets

In this case, the mass of the pellets is 5.50 g, which is equal to 0.00550 kg, and the velocity is 11.4 m/s.

Substituting these values into the kinetic energy equation, we have:
KE = (1/2)(0.00550 kg)(11.4 m/s)² = 0.35601 J

Since the kinetic energy is equal to the work done by the spring, which is also equal to the force exerted by the spring multiplied by the displacement, we can write:
KE = Fx

Rearranging the equation, we have:
F = KE / x

Substituting the values we have:
F = 0.35601 J / 0.0700 m = 5.08586 N

Now, we can substitute the force (F) and the displacement/compression (x) into Hooke's Law equation to find the spring constant (k).

5.08586 N = -k(0.0700 m)

Rearranging the equation to solve for k:
k = -5.08586 N / 0.0700 m

Calculating the value gives us:
k ≈ -72.655 N/m

However, the negative sign indicates that the force exerted by the spring is in the opposite direction to the compression. To obtain a positive spring constant value, we disregard the negative sign.

Therefore, the spring constant for the spring in the toy gun is approximately 72.655 N/m.