A 0.0122-kg bullet is fired straight up at a falling wooden block that has a mass of 2.17 kg. The bullet has a speed of 813 m/s when it strikes the block. The block originally was dropped from rest from the top of a building and had been falling for a time t when the collision with the bullet occurs. As a result of the collision, the block (with the bullet in it) reverses direction, rises, and comes to a momentary halt at the top of the building. Find the time t.

To find the time t, we can use the principle of conservation of momentum. When the bullet strikes the wooden block, momentum is conserved in the collision.

The momentum of an object is calculated by multiplying its mass with its velocity. In this case, the momentum of the bullet before the collision is equal to the momentum of the bullet and wooden block together after the collision.

Given:
Mass of the bullet, m1 = 0.0122 kg
Initial velocity of the bullet, u1 = 813 m/s
Mass of the wooden block, m2 = 2.17 kg
Final velocity of the bullet and wooden block together, v2 = 0 m/s (since it comes to a momentary halt at the top)

Using the conservation of momentum principle, we have:
(m1 * u1) = ((m1 + m2) * v2)

Now we can substitute the values into the equation and solve for v2:
(0.0122 kg * 813 m/s) = ((0.0122 kg + 2.17 kg) * 0 m/s)

Calculating further:
0.0099306 kg*m/s = 0.02662 kg * 0 m/s
0.0099306 kg*m/s = 0

Since 0 = 0.0099306 kg*m/s, this implies that the equation is satisfied regardless of the value of v2. Therefore, we cannot determine the exact time t from this equation alone.

However, we can make another observation. The block was initially dropped from rest from the top of a building, meaning its initial velocity was 0 m/s. Therefore, at time t, the block would have fallen a distance h which can be calculated using the formula:

h = (1/2) * g * t^2

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Since we want the block to reach its initial height again after the collision, the distance it has fallen should be equal to the height of the building. Using this information, we can set up the following equation:

h = (1/2) * g * t^2
h = 0.5 * 9.8 * t^2

Given:
h = height of the building

Now, we can solve for t by rearranging the equation:

t^2 = (2 * h) / g
t = sqrt((2 * h) / g)

Substituting the value of h (height of the building), you can calculate the value of t using the above equation.