A 23- bullet traveling 230 penetrates a 3.6 block of wood and emerges cleanly at 170 . If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges?
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, provided there are no external forces acting.
1. First, let's calculate the initial momentum of the bullet:
Initial momentum = mass × velocity
Given that the mass of the bullet is 23 g (0.023 kg) and the velocity is 230 m/s, the initial momentum can be calculated as follows:
Initial momentum = 0.023 kg × 230 m/s = 5.29 kg·m/s
2. Next, let's calculate the final momentum of the bullet:
Final momentum = mass × velocity
Given that the mass of the bullet is still 23 g (0.023 kg), but the velocity is 170 m/s, the final momentum can be calculated as follows:
Final momentum = 0.023 kg × 170 m/s = 3.91 kg·m/s
3. Since the block is initially stationary and there are no external forces acting, the total momentum after the event must be equal to the final momentum of the bullet.
4. Let's assume the mass of the wooden block is M kg and its velocity after the bullet emerges is V m/s. Then, the final momentum of the block can be calculated as follows:
Final momentum of the block = M × V
5. Now, we can equate the final momentum of the bullet to the final momentum of the block:
Final momentum of the bullet = Final momentum of the block
3.91 kg·m/s = M × V
6. Lastly, we need to determine the value of M using the given information. The bullet travels through 3.6 cm of wood, which can be considered as the block's thickness. Using the density formula, density = mass/volume, and the fact that the density of wood is approximately 700 kg/m^3, we can calculate the mass of the block:
Mass of the block = density × volume
Given that the block's volume is (3.6 cm)^3 = (0.036 m)^3 = 0.000046656 m^3, the mass of the block can be calculated as follows:
Mass of the block = 700 kg/m^3 × 0.000046656 m^3 = 0.0326 kg
7. Now that we know the mass of the block (M = 0.0326 kg) and the final momentum of the bullet (3.91 kg·m/s), we can substitute these values into the equation in step 5 and solve for V:
3.91 kg·m/s = 0.0326 kg × V
V = 3.91 kg·m/s / 0.0326 kg
V ≈ 120.09 m/s
Therefore, the velocity of the wooden block is approximately 120.09 m/s after the bullet emerges.
To find out the speed at which the wooden block moves after the bullet emerges, we can make use of the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, assuming no external forces are acting on the system.
Momentum is the product of an object's mass and its velocity. So, to solve this problem, we need to calculate the momentum of the bullet before it hits the block, as well as the momentum of the block and the bullet after the impact. Here's how we can do it step by step:
1. Calculate the momentum of the bullet before it hits the block:
Momentum (p) = Mass (m) × Velocity (v)
Given:
Mass of the bullet (m1) = 23 g = 0.023 kg
Velocity of the bullet (v1) = 230 m/s
Momentum of the bullet before impact (p1) = m1 × v1
2. Calculate the momentum of the block and the bullet after the impact:
Given:
Mass of the block (m2) = 3.6 kg
Velocity of the bullet after impact (v2) = 170 m/s (emerges cleanly)
Let's assume the velocity of the block after impact is (v3) m/s.
Momentum of the bullet after impact (p2) = m1 × v2
Momentum of the block after impact (p3) = m2 × v3
3. Apply the principle of conservation of momentum:
According to the principle, the total momentum before impact (p1) is equal to the total momentum after impact (p2 + p3):
p1 = p2 + p3
m1 × v1 = (m1 × v2) + (m2 × v3)
Now, we can solve this equation to find v3, the velocity of the block after the bullet emerges.
By substituting the given values and solving the equation, we can find the velocity (v3) at which the block moves after the bullet emerges.
initial momentum = final momentum
23 * 230 = 23 * 170 + 3.6 * v
solve for v