A 10.5 g bullet is fired into a 104 g wooden block initially at rest on a horizontal surface.The acceleration of gravity is 9.8 m/s2. After impact, the block slides 6.66 m before coming to rest.If the coefficient of friction between block and surface is 0.658 , what was the speed of the bullet immediately before impact?
Answer in units of m/s.
10.5 u = (10.5+104) v
F = m a = -.658 m g
so
a = -.658(g)
work done = F d
= m (.658)(g) (6.66)
work done = Ke after collision
so
(1/2) m v^2 = m(.658)(g)(6.66)
v^2 = 2(.658)(g)(6.66)
solve for v
then
u = (10.5+104) v/10.5
To find the speed of the bullet immediately before impact, we can use the principle of conservation of momentum. This principle states that the total momentum before the collision is equal to the total momentum after the collision.
Initially, the wooden block is at rest, so its momentum is zero. Therefore, the total momentum before the collision is equal to the momentum of the bullet:
Momentum before collision = mass of bullet × velocity of bullet before collision
After the collision, both the bullet and the wooden block will have a combined momentum. The bullet will penetrate the block, so its momentum is transferred to the block, causing it to move.
The total momentum after the collision is equal to the momentum of the block:
Momentum after collision = mass of block × velocity of block after collision
Since momentum is conserved, we can equate the two expressions:
mass of bullet × velocity of bullet before collision = mass of block × velocity of block after collision
Now, we need to find the velocity of the block after the collision. The block comes to rest after sliding a distance of 6.66 m on the horizontal surface. This means that friction acts on the block to bring it to a stop. The friction force can be calculated using the coefficient of friction:
Friction force = coefficient of friction × normal force
The normal force is equal to the weight of the block:
Normal force = mass of block × acceleration due to gravity
The work done by friction is equal to the force of friction times the distance over which it acts:
Work done by friction = Friction force × distance
The work done by friction is equal to the change in kinetic energy:
Work done by friction = (1/2) × mass of block × (velocity of block after collision)^2
Since the block comes to rest, its final velocity is zero. Therefore, we can solve for the initial velocity of the block:
(1/2) × mass of block × (velocity of block after collision)^2 = Friction force × distance
(1/2) × mass of block × (velocity of block after collision)^2 = coefficient of friction × mass of block × acceleration due to gravity × distance
Solving for the velocity of the block after the collision:
(velocity of block after collision)^2 = 2 × coefficient of friction × acceleration due to gravity × distance
velocity of block after collision = sqrt(2 × coefficient of friction × acceleration due to gravity × distance)
Now that we have the velocity of the block after the collision, we can substitute it back into the conservation of momentum equation:
mass of bullet × velocity of bullet before collision = mass of block × velocity of block after collision
velocity of bullet before collision = (mass of block × velocity of block after collision) / mass of bullet
Finally, substituting the given values into the equation will give us the speed of the bullet immediately before impact.