Question

If the spring of a jack-in-the-box is compressed a distance of 8.76 cm from its relaxed length and then released, what is the speed (in m/s) of the toy head when the spring returns to its natural length? Please consider the following assumptions:

The mass of the toy head is 61.8 g.
The spring constant is 37.4 N/m.
The toy head moves only in the vertical direction.

Answer 1

The speed of the toy head is approximately 0.872 m/s.

Step-by-step explanation:

To find the speed of the toy head when the spring returns to its natural length, we can use the principles of conservation of mechanical energy. When the spring is compressed, it gains potential energy. This potential energy is then converted to kinetic energy as the spring expands.

We can calculate the potential energy stored in the spring using the formula PE = 0.5 * k * x², where k is the spring constant and x is the compression distance. Substituting the given values, we get PE = 0.5 * 37.4 N/m * (0.0876 m)² = 0.1434 J.

Since the potential energy is converted to kinetic energy, we can equate the two using the formula KE = 0.5 * m * v², where m is the mass of the toy head and v is its velocity. Rearranging the formula, we get v = √(2 * KE / m).

Substituting the values, v = √(2 * 0.1434 J / 0.0618 kg) = 0.872 m/s.