A 12.0-g bullet is fired into a 1 100-g block of wood which is suspended as a ballistic pendulum. The combined mass swings up to a height of 8.50 cm. What was the kinetic energy of the combined mass immediately after the collision?
To calculate the kinetic energy of the combined mass immediately after the collision, we can use the principle of conservation of momentum.
First, let's identify all the given values:
Mass of the bullet (m₁) = 12.0 g = 0.012 kg
Mass of the block of wood (m₂) = 1100 g = 1.1 kg
Height reached by the combined mass (h) = 8.50 cm = 0.085 m
The mass of the combined system is the sum of the masses of the bullet and the block of wood:
m = m₁ + m₂ = 0.012 kg + 1.1 kg = 1.112 kg
When the bullet is fired into the block of wood, it embeds itself in the wood, and the two masses move together as a single unit. The initial momentum (pᵢ) of the system is equal to the momentum of the bullet before the collision, which is given by:
pᵢ = m₁ * v₁
where v₁ is the initial velocity of the bullet.
Since the bullet is embedded in the wood, the bullet and the wood follow a common vertical path during their upward swing. At the maximum height (H) reached by the combined system, all the initial kinetic energy is converted into potential energy.
Using the principle of conservation of mechanical energy, we can write:
Initial kinetic energy (KEᵢ) = Potential energy at maximum height (PEₘₐₓ)
The kinetic energy of the combined mass immediately after the collision is given by:
KEᵢ = 1/2 * m * v²
where v is the velocity of the combined mass immediately after the collision.
To solve for v, we need the height (H) at which the maximum potential energy is achieved. Since it is given that the combined mass swings up to a height of 8.50 cm, we can use this information to find the value of H.
The maximum potential energy (PEₘₐₓ) is given by:
PEₘₐₓ = m * g * H
where g is the acceleration due to gravity (approximately 9.8 m/s²).
Now we have two equations:
Initial kinetic energy (KEᵢ) = PEₘₐₓ
1/2 * m * v² = m * g * H
We can rearrange the second equation to solve for v:
v = sqrt(2 * g * H)
Substituting the given values, we have:
v = sqrt(2 * 9.8 m/s² * 0.085 m)
Solving for v gives:
v ≈ 1.836 m/s
Finally, we can find the initial kinetic energy (KEᵢ) using the equation:
KEᵢ = 1/2 * m * v²
Substituting the values:
KEᵢ = 1/2 * 1.112 kg * (1.836 m/s)²
Evaluating this expression gives the result:
KEᵢ ≈ 1.991 J (rounded to three significant figures)
Therefore, the kinetic energy of the combined mass immediately after the collision is approximately 1.991 Joules.