A horizontal spring with k = 70 N/m has one end attached to a wall and the other end free. An 95 g wad of putty is thrown horizontally at 3.4 m/s directly toward the free end.

Find the maximum spring compression.

Maximum spring compression is obtained when (1/2) k X^2 equals the initial kinetic energy of the wad.

(1/2)k X^2 = (1/2)M V^2

X = V sqrt(M/k)

convert grams into kilo grams:

95/1000 = .095 kg
x = sqrt MV^2/k
x = sqrt .095(3.4)^2/70
x = .12525
x = .13 m
x = 13 cm

To find the maximum spring compression, we need to use the conservation of energy principle. The potential energy stored in the spring is equal to the kinetic energy of the putty when it reaches the spring.

First, let's find the kinetic energy of the putty. The formula for kinetic energy is:

Kinetic energy = 1/2 * mass * velocity ^2

Plugging in the values given:

Mass = 95 g = 0.095 kg
Velocity = 3.4 m/s

Kinetic energy = 1/2 * 0.095 kg * (3.4 m/s) ^ 2

Next, we equate the kinetic energy to the potential energy stored in the spring. The formula for potential energy in a spring is:

Potential energy = 1/2 * spring constant * compression ^2

Since the spring is horizontal, there is no gravitational potential energy. Therefore, the total mechanical energy is equal to the potential energy stored in the spring.

Setting the kinetic energy equal to the potential energy:

1/2 * 0.095 kg * (3.4 m/s) ^ 2 = 1/2 * 70 N/m * compression ^2

Simplifying:

0.5 * 0.095 kg * 11.56 m^2/s^2 = 0.5 * 70 N/m * compression ^2

Dividing both sides by 0.5:

0.095 kg * 11.56 m^2/s^2 = 70 N/m * compression ^2

0.5474 kg·m^2/s^2 = 70 N/m * compression ^2

Now, we can solve for the compression by rearranging the equation:

compression ^2 = 0.5474 kg·m^2/s^2 / (70 N/m)

compression ^2 = 0.00782 kg·m / N

Taking the square root of both sides:

compression = sqrt(0.00782 kg·m/N)

compression ≈ 0.088 m

Therefore, the maximum spring compression is approximately 0.088 meters.