A 25 g bullet is fired from a gun into a 1.35 kg block of wood initially at rest. (The bullet sticks in the wood). The wood is driven 9.5 m across a horizontal surface with coefficient of friciton 0.25. Find the speed of the bullet just before impact.
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To find the speed of the bullet just before impact, we can use the Law of Conservation of Momentum.
The Law of Conservation of Momentum states that the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting on the system.
Before impact, we have two objects: the bullet and the block of wood. Let's denote the initial velocity of the bullet as "v" and the final velocity of the bullet and wood combination as "vf".
The momentum before impact is given by the product of the mass of the bullet (mb) and its initial velocity (v):
Momentum before impact = mb * v
The momentum after impact is given by the combined mass of the bullet and wood (mb + mw) multiplied by the final velocity (vf):
Momentum after impact = (mb + mw) * vf
According to the Law of Conservation of Momentum, these two quantities are equal:
mb * v = (mb + mw) * vf
We are given the mass of the bullet (mb = 0.025 kg), the mass of the wood (mw = 1.35 kg), and the coefficient of friction (μ = 0.25). However, we need to find the final velocity vf to determine the speed of the bullet just before impact.
To find vf, we need to consider the forces acting on the wood. The force acting on the wood is the force of friction, which can be calculated using the equation:
Frictional force = coefficient of friction * normal force
The normal force (N) is equal to the weight of the wood, which can be calculated by multiplying its mass by the acceleration due to gravity (g ≈ 9.8 m/s^2):
Normal force = mw * g
The frictional force can be found by substituting these values into the equation:
Frictional force = μ * (mw * g)
Since the frictional force acts in the opposite direction to the motion, it will cause a deceleration of the wood. We can calculate this deceleration using the equation:
Deceleration = Frictional force / (mb + mw)
We can now use this deceleration to determine the final velocity vf of the bullet and wood combination using the equation of motion:
vf^2 = v^2 + 2 * acceleration * displacement
In this case, the displacement can be taken as the distance the wood is driven across the horizontal surface, which is given as 9.5 m.
Once we have the final velocity vf, we know that it represents the combination of the bullet and wood. Therefore, to find the speed of the bullet just before impact, we need to find the velocity of the bullet within the final velocity.
I will now perform the calculations to find the final velocity vf and then extract the speed of the bullet.