An 8.00 g bullet is fired into a 160 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 = 1.6 m from the bottom of the table. Determine the initial speed of the bullet.
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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 of a system of objects remains constant, provided no external forces are acting on the system.
In this case, we can consider the system consisting of the bullet and the block. Before the collision, the bullet is moving and the block is at rest. After the collision, the bullet is embedded in the block, and the combined system continues to move together until it reaches the ground.
The momentum of an object can be calculated by multiplying its mass by its velocity. So, we can write the conservation of momentum equation as:
(mass of bullet) × (velocity of bullet) = (mass of bullet + mass of block) × (velocity of combined system)
Let's assign variables to the given values:
Mass of bullet (m1) = 8.00 g = 0.008 kg
Mass of block (m2) = 160 g = 0.160 kg
Height of table (h) = 1.00 m
Distance from the bottom of the table to where the block lands (d) = 1.6 m
We need to find the velocity of the bullet (v1).
The first step is to determine the velocity of the combined system (v2) when it reaches the ground. We can use the equation of motion to calculate this.
Using the equation:
d = (initial velocity × time) + (1/2) × (acceleration × time^2)
Since the block is initially at rest, its initial velocity is 0 m/s. We can rewrite the equation as:
d = (1/2) × (acceleration × time^2)
Since the only acceleration acting on the block is due to gravity (g), we can substitute that:
d = (1/2) × (g × time^2)
Rearranging the equation to solve for time:
time = sqrt((2d)/g)
Substituting the given values:
time = sqrt((2 × 1.6 m)/(9.8 m/s^2)) = 0.6449 s (approximately)
Now, we can substitute the values into the conservation of momentum equation:
(0.008 kg) × (v1) = (0.008 kg + 0.160 kg) × (v2)
Simplifying:
0.008 kg × v1 = 0.168 kg × v2
Now, we can use the equation of motion to calculate the velocity of the combined system:
v2 = g × time = 9.8 m/s^2 × 0.6449 s = 6.3279 m/s (approximately)
Substituting the value of v2 into the conservation of momentum equation:
0.008 kg × v1 = 0.168 kg × 6.3279 m/s
Simplifying further:
v1 = (0.168 kg × 6.3279 m/s) / 0.008 kg
v1 = 1.3258 m/s (approximately)
Therefore, the initial speed of the bullet is approximately 1.3258 m/s.