a) A bullet (m = 0.0250 kg) is fired with a speed of 91.00 m/s and hits a block (M = 2.80 kg) supported by two light strings as shown, stopping quickly. Find the height to which the block rises.
b)Find the angle (in degrees) through which the block rises, if the strings are 0.360 m in length.
To solve this problem, we can use principles of conservation of momentum and conservation of energy. Here's how you can find the solution to each part of the question:
a) To find the height to which the block rises, we can first calculate the initial momentum (momentum before the collision) and the final momentum (momentum after the collision) of the bullet and the block.
1. Calculate the initial momentum:
The initial momentum of the bullet is given by the formula: p = m*v, where m is the mass of the bullet and v is its velocity.
Initial momentum of the bullet = (0.0250 kg) * (91.00 m/s) = 2.275 kg⋅m/s
2. Calculate the final momentum:
Since the bullet stops quickly upon hitting the block, its final momentum is zero (because its velocity becomes zero after the collision).
Final momentum of the bullet = 0 kg⋅m/s
3. Apply the principle of conservation of momentum:
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Initial momentum of the bullet = Final momentum of the bullet + Final momentum of the block
2.275 kg⋅m/s = 0 kg⋅m/s + Final momentum of the block
Therefore, Final momentum of the block = 2.275 kg⋅m/s
4. Calculate the change in momentum of the block:
Change in momentum of the block can be calculated using the formula: Change in momentum = Final momentum - Initial momentum.
Change in momentum of the block = Final momentum of the block - 0 kg⋅m/s = 2.275 kg⋅m/s
5. Calculate the velocity of the block:
The velocity of the block can be determined by using the formula: Final momentum = m*v, where m is the mass of the block and v is its velocity.
2.275 kg⋅m/s = (2.80 kg) * v
v = 2.275 kg⋅m/s / (2.80 kg) ≈ 0.813 m/s
6. Calculate the height to which the block rises:
To find the height to which the block rises, we can consider the principle of conservation of mechanical energy. The initial mechanical energy (kinetic energy of the bullet) is completely converted into potential energy (height of the block).
The potential energy gained by the block can be calculated using the formula: Potential energy = mass * acceleration due to gravity * height
Potential energy = (2.80 kg) * (9.81 m/s^2) * height
The initial mechanical energy of the bullet is given by the formula: Initial kinetic energy = 1/2 * mass * velocity^2
Initial kinetic energy = 1/2 * (0.0250 kg) * (91.00 m/s)^2
According to the principle of conservation of energy, we can equate the initial kinetic energy to the potential energy gained by the block:
1/2 * (0.0250 kg) * (91.00 m/s)^2 = (2.80 kg) * (9.81 m/s^2) * height
Now, solve the equation for height:
height = (1/2 * (0.0250 kg) * (91.00 m/s)^2) / ((2.80 kg) * (9.81 m/s^2))
= 0.093 m
Therefore, the height to which the block rises is approximately 0.093 meters.
b) To find the angle through which the block rises, we can consider the geometry of the situation.
The strings supporting the block form a triangle with the vertical. The length of each string is given as 0.360 m. Let's consider this triangle:
/|
/ |
/ | h
/ |
/____|
L1 L2
Here, h represents the height to which the block rises. L1 and L2 represent the lengths of the strings.
We can find the angle θ (in degrees) through which the block rises by using the formula: cos(θ) = h / L1.
Rearranging the formula, we get:
θ = cos^(-1)(h / L1)
Substituting the calculated value of h = 0.093 m and L1 = 0.360 m into the equation, we can find the angle θ.
θ = cos^(-1)(0.093 m / 0.360 m)
θ ≈ 73.86 degrees
Therefore, the angle through which the block rises is approximately 73.86 degrees.