1. Calculate the Elastic Potential Energy
* The elastic potential energy stored in the rubber band is given by:
* PE = (1/2) * k * x²
* Where:
* PE is the potential energy
* k is the spring constant of the rubber band (we'll need to find this)
* x is the distance the rubber band is stretched (0.15 m)
2. Determine the Spring Constant (k)
* We can find the spring constant using the force applied and the distance stretched:
* F = k * x
* k = F / x = 27 N / 0.15 m = 180 N/m
3. Calculate the Kinetic Energy
* Assuming no energy loss due to friction or other factors, the elastic potential energy is converted into kinetic energy of the stone:
* KE = (1/2) * m * v²
* Where:
* KE is the kinetic energy
* m is the mass of the stone (0.025 kg)
* v is the initial speed of the stone (what we want to find)
4. Equate Potential and Kinetic Energy
* Since energy is conserved:
* PE = KE
* (1/2) * k * x² = (1/2) * m * v²
5. Solve for the Initial Speed (v)
* Substitute the known values and solve for v:
* (1/2) * 180 N/m * (0.15 m)² = (1/2) * 0.025 kg * v²
* v² = (180 N/m * 0.15 m²) / 0.025 kg
* v = √((180 * 0.15²) / 0.025) ≈ 11.0 m/s
Therefore, the initial speed of the stone is approximately 11.0 m/s.
Important Note: This calculation makes several assumptions, including:
* No energy loss: In reality, there will be some energy loss due to friction, air resistance, and the imperfect elasticity of the rubber band.
* Ideal spring behavior: We are assuming the rubber band acts like a perfect spring, which might not be entirely accurate.
These factors mean the actual initial speed of the stone will likely be slightly lower than the calculated value.
