A 5.88 g bullet is fired vertically into a 0.9 kg block of wood. The bullet gets stuck in the block, and the impact lifts the block 0.87 m up. (That is, the
block — with the bullet stuck in it — rises 0.87 m up above its initial position, and then falls back down.)
Given g = 9.8 m/s2. What was the initial velocity of the bullet?
Answer in units of m/s.
the KE of the block/bullet must be equal to the change in PE of the block /bullet
1/2 (M+m)v1^2=(m+M)gh
solve for V1^2, the initial velocity of the block/bullet combination.
but at impact of the bullet, conservation of momentum applies
mVbullet= (m+M)V1
solve for vbullet.
Well, it sounds like the bullet really made an impression on that block, quite literally! To find the initial velocity of the bullet, we can make use of the principle of conservation of momentum.
First, let's find the momentum of the bullet before it hits the block. We can use the equation p = mv, where p is momentum, m is mass, and v is velocity. Given that the bullet has a mass of 5.88 g (or 0.00588 kg), we can say its initial momentum is p_bullet = (0.00588 kg)(v_bullet).
Now, let's consider the momentum of the block after the bullet is lodged in it. The momentum of the block can be calculated as p_block = (0.9 kg)(v_block), where v_block is the upward velocity of the block.
According to the conservation of momentum, the total momentum before the impact should be equal to the total momentum after. So we can write the equation:
p_bullet = p_block
(0.00588 kg)(v_bullet) = (0.9 kg)(v_block)
Now, the bullet gets stuck in the block, so the velocity of the block when it's at its peak (0.87 m above its initial position) should be zero. At this point, all the kinetic energy has been converted to potential energy. Therefore, we can use the equation:
mgh = (1/2)mv^2
where m is mass, g is gravitational acceleration, h is the height, and v is velocity. Rearranging the equation, we get:
v_block = sqrt(2gh)
Plugging this into our previous equation, we have:
(0.00588 kg)(v_bullet) = (0.9 kg)(sqrt(2gh))
Now, let's substitute the values we know. The gravitational acceleration, g, is 9.8 m/s^2, and the height, h, is 0.87 m:
(0.00588 kg)(v_bullet) = (0.9 kg)(sqrt(2(9.8 m/s^2)(0.87 m)))
Simplifying this equation will give us the initial velocity of the bullet. Calculating that... drumroll, please...
*Calculating...*
*Calculating...*
*Blinking lights...*
The initial velocity of the bullet is approximately *tada!* 43.27 m/s.
So, there you have it! The bullet must have had quite the speed to lift that block up! Keep in mind that these calculations assume no loss of energy due to friction or air resistance.
To find the initial velocity of the bullet, we can use the principle of conservation of momentum.
Step 1: Find the change in momentum of the system.
The momentum before the bullet gets stuck in the block is given by:
initial momentum = mass of the bullet * initial velocity of the bullet
The momentum after the bullet gets stuck in the block is given by:
final momentum = total mass (bullet + block) * final velocity of the system
Since the bullet gets stuck in the block, the final velocity of the system is 0, as it momentarily comes to rest.
The change in momentum is then:
change in momentum = final momentum - initial momentum
Step 2: Calculate the initial momentum of the system.
The total mass of the system is the mass of the bullet (5.88 g) plus the mass of the block (0.9 kg converted to grams):
total mass = mass of the bullet + mass of the block
Step 3: Solve for the initial velocity of the bullet.
Using the equation for change in momentum, we have:
change in momentum = (total mass) * (final velocity) - (mass of the bullet) * (initial velocity of the bullet)
Since the final velocity is 0 (the block momentarily comes to rest), the equation becomes:
change in momentum = - (mass of the bullet) * (initial velocity of the bullet)
Now we can solve for the initial velocity of the bullet:
initial velocity of the bullet = change in momentum / mass of the bullet
Step 4: Calculate the change in momentum.
The change in momentum is equal to the momentum before the bullet gets stuck (initial momentum), because the final momentum is 0:
change in momentum = initial momentum
Plugging in the values, we have:
initial velocity of the bullet = (initial momentum) / (mass of the bullet) = (initial momentum) / (5.88 g)
Step 5: Convert units and calculate.
To calculate the initial velocity of the bullet, we need to convert grams to kilograms to match the units of mass in the equation. So, we divide the mass of the bullet by 1000 to convert to kg:
mass of the bullet = 5.88 g / 1000 = 0.00588 kg
Now we can calculate the initial velocity of the bullet using the equation:
initial velocity of the bullet = (initial momentum) / (mass of the bullet)
Since momentum is calculated by multiplying mass and velocity, we can restate this equation as:
initial velocity of the bullet = (mass of the bullet * initial velocity of the bullet) / (mass of the bullet)
Simplifying further:
initial velocity of the bullet = initial velocity of the bullet
Therefore, the initial velocity of the bullet remains unknown with the information given in the problem.
To find the initial velocity of the bullet, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
In this case, we have a bullet of mass 5.88 g and an unknown initial velocity, colliding with a block of wood weighing 0.9 kg. Let's assume the initial velocity of the bullet is V.
The momentum of the bullet before the collision is given by (mass of the bullet) * (initial velocity of the bullet), which is (5.88 g) * (V).
The momentum of the block of wood, after the collision, while it is rising up is given by (mass of the block) * (final velocity), which is (0.9 kg) * (0 m/s) since the block momentarily stops at its highest point.
The momentum of the bullet and the block together, after the collision, but before they separate is given by (mass of the bullet + mass of the block) * (final velocity), which is (5.88 g + 0.9 kg) * (0 m/s) since the bullet and the block move together.
According to the conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision:
(5.88 g) * (V) = (0.9 kg) * (0 m/s) + (5.88 g + 0.9 kg) * (0 m/s)
Simplifying the equation, we get:
(5.88 g) * (V) = 0
Dividing both sides of the equation by (5.88 g), we find:
V = 0
Therefore, the initial velocity of the bullet is 0 m/s.