A bullet of mass 11.2 g is fired into an initially stationary block and comes to rest in the block. The block, of mass 1.01 kg, is subject to no horizontal external forces during the collision with the bullet. After the collision, the block is observed to move at a speed of 4.40 m/s.
(a) Find the initial speed of the bullet.
(b) How much kinetic energy is lost?
m1 =0.0112 kg, v =?
m2 = 1.01kg, u = 4.4 m/s
The law of conservation of linear momentum
m1•v + 0 = (m1+m2) •u,
(a) v =(m1+m2) •u/m1.
(b) ΔKE = m1•v^2/2 – (m1+m2) •^2/2.
To find the initial speed 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.
The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum = mass * velocity.
Let's denote the initial velocity of the bullet as v and the final velocity of the block as V.
Before the collision:
Momentum of bullet = mass of bullet * initial velocity of bullet = (11.2 g) * v
Momentum of block = mass of block * 0 (initial velocity of block is zero since it is initially stationary) = 0
After the collision:
Momentum of bullet = mass of bullet * 0 (bullet comes to rest in the block, so its final velocity is zero) = 0
Momentum of block = mass of block * final velocity of block = (1.01 kg) * 4.40 m/s
According to the conservation of momentum principle, the total momentum before the collision is equal to the total momentum after the collision. Therefore:
(11.2 g) * v + 0 = 0 + (1.01 kg) * 4.40 m/s.
Now, we need to convert the mass and velocity of the bullet to SI units (kilograms and meters per second, respectively). 1 g is equal to 0.001 kg.
(0.0112 kg) * v = (1.01 kg) * 4.40 m/s.
Simplifying the equation:
v = (1.01 kg * 4.40 m/s) / (0.0112 kg)
v ≈ 399.1 m/s
Therefore, the initial speed of the bullet is approximately 399.1 m/s.
To find the amount of kinetic energy lost, we can use the principle of conservation of mechanical energy. The initial total kinetic energy is equal to the sum of the kinetic energy of the bullet and the block before the collision, while the final total kinetic energy is equal to the kinetic energy of the block after the collision.
The kinetic energy of an object is defined as one-half the product of its mass and the square of its velocity. Mathematically, kinetic energy = (1/2) * mass * velocity^2.
Let's denote the initial kinetic energy of the bullet as K1, the initial kinetic energy of the block as K2, and the final kinetic energy of the block as K3.
Before the collision:
K1 = (1/2) * mass of bullet * initial velocity of bullet^2
K2 = (1/2) * mass of block * 0^2 = 0
After the collision:
K3 = (1/2) * mass of block * final velocity of block^2 = (1/2) * (1.01 kg) * (4.40 m/s)^2
According to the conservation of mechanical energy principle, the initial total kinetic energy is equal to the final total kinetic energy. Therefore:
K1 + K2 = K3
(1/2) * mass of bullet * initial velocity of bullet^2 + 0 = (1/2) * (1.01 kg) * (4.40 m/s)^2.
Substituting the values and simplifying:
(1/2) * (0.0112 kg) * (v^2) = (1/2) * (1.01 kg) * (4.40 m/s)^2.
Now, we'll substitute the value of v we found in part (a):
(1/2) * (0.0112 kg) * (399.1 m/s)^2 = (1/2) * (1.01 kg) * (4.40 m/s)^2.
Calculating:
(1/2) * (0.0112 kg) * (399.1 m/s)^2 ≈ 894.2 J
Therefore, the amount of kinetic energy lost during the collision is approximately 894.2 J.