An 7.39-g bullet is shot into a 4.00-kg block, at rest on a frictionless horizontal surface. The bullet remains lodged in the block.The block moves into a spring and compresses it by 15.1 cm before stopping. The force constant of the spring is 1,842 N/m.What was the initial velocity of the bullet?Express your answer in m/s.
Question
An 7.39-g bullet is shot into a 4.00-kg block, at rest on a frictionless horizontal surface. The bullet remains lodged in the block.The block moves into a spring and compresses it by 15.1 cm before stopping. The force constant of the spring is 1,842 N/m.What was the initial velocity of the bullet?Express your answer in m/s.
Solution
To find the initial velocity of the bullet, we can use the principle of conservation of momentum.
Step 1: Calculate the momentum of the bullet before it hits the block. The momentum of an object is given by the equation p = mv, where p is the momentum, m is the mass, and v is the velocity. Given that the mass of the bullet is 7.39 g (or 0.00739 kg) and the bullet is at rest, the initial momentum of the bullet is 0 kg*m/s.
Step 2: Calculate the momentum of the block and bullet system after the collision. Since the bullet remains lodged in the block, the mass of the block and bullet system is the sum of their masses, which is 4.00 kg + 0.00739 kg = 4.00739 kg. Let's assume the final velocity of the block and bullet system is v_f. Using the principle of conservation of momentum, the momentum before the collision is equal to the momentum after the collision. Therefore, 0 kg*m/s = (4.00739 kg) * v_f.
Step 3: Calculate the velocity of the block and bullet system after the collision. Solving the equation from step 2 for v_f, we get v_f = 0 m/s.
Step 4: Calculate the work done by the spring. The work done by the spring is given by the equation W = (1/2)kx^2, where W is the work done, k is the force constant of the spring, and x is the compression of the spring. Given that the force constant of the spring is 1,842 N/m and the spring is compressed by 15.1 cm (or 0.151 m), the work done by the spring is W = (1/2)(1,842 N/m)(0.151 m)^2.
Step 5: Calculate the change in kinetic energy of the block and bullet system. The change in kinetic energy is equal to the work done by the spring. Therefore, the change in kinetic energy is equal to the work done by the spring calculated in step 4.
Step 6: Calculate the initial kinetic energy of the block and bullet system. The initial kinetic energy of the block and bullet system is given by the equation KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity. Since the block and bullet system is initially at rest, the initial kinetic energy is 0 J.
Step 7: Set up the equation for the conservation of mechanical energy. The conservation of mechanical energy states that the initial kinetic energy plus the work done by external forces is equal to the final kinetic energy. Therefore, 0 J + (work done by the spring) = (change in kinetic energy).
Step 8: Solve the equation from step 7 for the change in kinetic energy. Substituting the values from step 5 into the equation from step 7, we get (change in kinetic energy) = (work done by the spring).
Step 9: Solve the equation from step 8 for the final kinetic energy. Since the final kinetic energy is equal to the change in kinetic energy, we have (final kinetic energy) = (work done by the spring).
Step 10: Set up the equation for the final kinetic energy. The final kinetic energy is given by the equation KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity. Substituting the values from step 3 into the equation, we get (final kinetic energy) = (1/2)(4.00739 kg)(0 m/s)^2.
Step 11: Solve the equation from step 10 for the final kinetic energy. Calculating the equation from step 10, we find that the final kinetic energy is 0 J.
Step 12: Set up the equation for the conservation of mechanical energy. Using the equation from step 7, we have 0 J + (work done by the spring) = 0 J.
Step 13: Solve the equation from step 12 for the work done by the spring. Since the work done by the spring is equal to 0 J, we have (work done by the spring) = 0 J.
Step 14: Calculate the initial kinetic energy of the block and bullet system. Using the equation from step 7, we have 0 J + 0 J = (change in kinetic energy). Therefore, the change in kinetic energy is 0 J.
Step 15: Set up the equation for the change in kinetic energy. The change in kinetic energy is given by the equation ΔKE = (1/2)mv^2 - (1/2)m(0 m/s)^2, where ΔKE is the change in kinetic energy, m is the mass, and v is the velocity. Substituting the values from step 6 into the equation, we get 0 J = (1/2)(4.00739 kg)v^2.
Step 16: Solve the equation from step 15 for the initial velocity of the bullet. Rearranging the equation from step 15, we have v^2 = 0 J / (1/2)(4.00739 kg). Simplifying the equation, we get v^2 = 0 m^2/s^2. Taking the square root of both sides, we find that the initial velocity of the bullet is v = 0 m/s.
Therefore, the initial velocity of the bullet is 0 m/s.
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