A 0.026 kg bullet is fired vertically at 205 m/s into a 0.15 kg baseball that is initially at rest. How high does the combined bullet and baseball rise after the collision, assuming the bullet embeds itself in the ball?
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To determine the height reached by the combined bullet and baseball after the collision, we can use the principle of conservation of momentum.
First, let's find the initial momentum of the bullet:
Initial momentum of the bullet (P₁) = mass of the bullet (m₁) × velocity of the bullet (v₁)
P₁ = 0.026 kg × 205 m/s
P₁ = 5.33 kg·m/s
Since the baseball is initially at rest, its initial momentum is zero:
Initial momentum of the baseball (P₂) = 0 kg·m/s
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
Total initial momentum = Total final momentum
P₁ + P₂ = final momentum
Since the bullet embeds itself in the baseball, the final momentum is the momentum of the combined bullet and baseball system after the collision.
Final momentum (P_f) = mass of the combined system (m_f) × final velocity of the combined system (v_f)
The mass of the combined system is the sum of the masses of the bullet and the baseball:
m_f = m₁ + m₂
m_f = 0.026 kg + 0.15 kg
m_f = 0.176 kg
Since the bullet embeds itself in the baseball, the final velocity of the combined system is zero.
Plugging these values into the momentum conservation equation:
P₁ + P₂ = m_f × v_f
5.33 kg·m/s + 0 kg·m/s = 0.176 kg × v_f
Solving for the final velocity (v_f):
v_f = (5.33 kg·m/s) / (0.176 kg)
v_f = 30.27 m/s
The final velocity is the upward velocity of the combined bullet and baseball system after the collision. To determine the maximum height reached, we can use the principle of conservation of mechanical energy.
The potential energy at the maximum height is equal to the initial kinetic energy of the system:
Potential energy at maximum height = (1/2) × (m₁ + m₂) × (v_f)²
Plugging in the values:
Potential energy at maximum height = (1/2) × (0.176 kg) × (30.27 m/s)²
Potential energy at maximum height ≈ 81.29 J
The potential energy at maximum height is equal to the gravitational potential energy at maximum height:
Potential Energy at maximum height = m₁gh
Solving for h, the height reached:
h = Potential Energy at maximum height / (m₁g)
The mass of the bullet, m₁, is 0.026 kg, and the acceleration due to gravity, g, is approximately 9.8 m/s².
Plugging in the values:
h = 81.29 J / (0.026 kg × 9.8 m/s²)
h ≈ 308.96 meters
Therefore, the combined bullet and baseball rise to a height of approximately 308.96 meters after the collision.