Final answer:
Using the conservation of energy principle, the velocity of a ball launched from a spring in a pinball machine can be calculated. The potential energy stored in the compressed spring is converted into the ball's kinetic energy, resulting in the ball traveling at approximately 2.13 meters per second after launch.
Step-by-step explanation:
To determine how quickly the ball is traveling after being launched from the pinball machine, we can use the principle of conservation of energy. The energy stored in the compressed spring is converted into kinetic energy of the ball when it's launched. The potential energy stored in the spring (PEs) when the spring is compressed is given by the formula:
PEs = ½ k x²,
where k is the spring constant and x is the compression distance of the spring. In this case, k = 93.4 N/m, and the spring is compressed by 0.08 m (0.1 m - 0.02 m).
The kinetic energy (KE) of the ball when it is launched is equal to the potential energy stored in the spring, assuming no energy is lost to friction or air resistance. Therefore:
KE = PEs
Since the kinetic energy of the ball is given by KE = ½ m v², where m is the mass of the ball and v is its velocity, we can equate the kinetic energy to the potential energy of the spring to solve for velocity:
½ m v² = ½ k x²
Solving for v gives us:
v = √((k x²) / m)
Plugging in the values, we get:
v = √((93.4 N/m × (0.08 m)²) / 0.134 kg)
Calculating this, we get:
v = √((93.4 × 0.0064) / 0.134)
v = √(0.59776 / 0.134)
v is approximately 2.13 m/s.
The ball is traveling at approximately 2.13 meters per second after being launched from the pinball machine.