Final answer:
To find the initial speed of the bullet, the conservation of energy principle is applied, where the kinetic energy of the bullet pre-collision is equal to the potential energy of the pendulum and bullet post-collision. Additional information, like the bullet's mass, is needed to calculate the exact initial speed.
Step-by-step explanation:
The student is asking about the initial speed of a bullet fired into a pendulum bob, which becomes embedded in it, causing the pendulum to rise a vertical distance post-collision. To solve for the initial speed of the bullet, we will use the principle of conservation of energy. Initially, the bullet has kinetic energy; after the collision, the bullet-bob combination has potential energy due to the height it has risen.
The potential energy (PE) at the peak of the pendulum's swing can be calculated using the formula PE = m*g*h, where m is the mass of the pendulum and bullet system, g is the acceleration due to gravity (approximately 9.81 m/s2), and h is the height risen (0.8m). The total mass is the mass of the pendulum (2.5 kg) plus the mass of the bullet, which we need to calculate. The kinetic energy (KE) of the bullet before the impact is KE = 0.5 * m * v2, where v is the velocity of the bullet.
Assuming no energy loss due to friction or air resistance (an idealized scenario), the kinetic energy of the bullet before the impact would equal the potential energy at the peak after the collision. Therefore, solving for v gives us the initial speed of the bullet:
- Calculate the potential energy at the peak: PE = m * g * h
- Assume the kinetic energy of the bullet before collision equals the potential energy at the peak: KE_initial = PE_peak
- Solve for the initial velocity of the bullet: v = sqrt((2*PE)/m)
For the actual numerical calculations, additional information, such as the mass of the bullet, is required.