An 8.00 g bullet is fired into a 250 g block that is initially at rest at the edge of a table of 1.00 m height. The bullet remains in the block, and after the impact the block lands d = 1.8 m from the bottom of the table. Determine the initial speed of the bullet.
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Calculate how long it takes the block to reach the floor.
1.00 m = (g/2) t^2
t = 0.452 s
Then calculate the horizontal velocity component as it leaves the edge of the table
Vx = 1.8 m/0.452s = 3.98 m/s
Then use conservation of momentum to get the velocity of the bullet, v.
m v = (m+M) Vx
m is the bullet mass and M is the block's mass.
v = (258/8) Vx = ___
To determine the initial speed of the bullet, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision, assuming no external forces act on the system.
The total momentum before the collision is given by the momentum of the bullet, which can be calculated using the formula:
P_bullet = m_bullet * v_bullet,
where m_bullet is the mass of the bullet and v_bullet is its velocity.
The total momentum after the collision is equal to the momentum of the combined bullet and block system, which can be calculated using the formula:
P_after = (m_bullet + m_block) * v_after,
where m_block is the mass of the block, v_after is the velocity of the combined bullet and block system after the collision.
Given:
m_bullet = 8.00 g = 0.008 kg,
m_block = 250 g = 0.250 kg,
d = 1.8 m.
Since the bullet remains in the block, we can say that the combined object moves together after the impact. Therefore, we can assume that the velocity of the combined bullet and block system after the collision (v_after) is the same.
Since the block is initially at rest, its initial momentum is zero. Hence, the total momentum before the collision is equal to the momentum of the bullet:
P_before = P_bullet = m_bullet * v_bullet.
The total momentum after the collision is given by:
P_after = (m_bullet + m_block) * v_after.
According to the principle of conservation of momentum, P_before = P_after:
m_bullet * v_bullet = (m_bullet + m_block) * v_after.
Substituting the given values into the equation, we can solve for v_bullet:
0.008 kg * v_bullet = (0.008 kg + 0.250 kg) * v_after.
Since the bullet remains in the block, the horizontal distance traveled by the combined bullet and block system (d) is equal to the horizontal distance traveled by the block (since the bullet's initial velocity is equal to the velocity of the block after the collision):
d = v_after * t,
where t is the time taken for the block to reach the horizontal distance d.
To find t, we can use the equation of motion:
d = (1/2) * g * t^2,
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Solving for t, we have:
t^2 = (2 * d) / g,
t = sqrt((2 * d) / g).
Substituting the given value of d into the equation, we can calculate t.
Finally, substituting the value of t into the equation m_bullet * v_bullet = (m_bullet + m_block) * v_after, we can solve for v_bullet, which is the initial speed of the bullet.