Liam fires a bullet into a block suspended by two cables, embedding the bullet in the block. The mass of the bullet is 5g and the mass of the block is 1g.
1) If the maximum height of the block is ,h, is 35 cm, how fast was the bullet originally moving?
(can you explain the steps i'm really confused)
m1= mass of the bullet
m2= mass of the block
Vf=sqrt(2gh)
v1= initial velocity of bullet
v2= initial velocity of the block = 0
so
first is conservation of momentum
m1v1+m2v2=(m1+v2)Vf
solve for v1 (note m2v2=0)
v1=((m1+m2)/m2)/Vf
to find Vf look at conservation of energy
KE+PE=KE'+PE'
.5(m1+m2)Vf^2+0=0+(m1+m2)gh
Solve for Vf and you get Vf=Sqrt(2gh)
now v1=((m1+m2)/m2)/(sqrt(2gh)) and solve
v1= 15.715 m/s
To solve this problem, we can make use of the principles of conservation of energy and momentum. Let's break down the steps to find the original speed of the bullet:
Step 1: Determine the potential energy of the block at its maximum height.
The potential energy (PE) of an object is given by the equation: PE = mgh, where m is the mass (in kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height (in meters).
In this case, the mass of the block is 1g (or 0.001 kg) and the height is given as 35 cm (or 0.35 m). So, we can calculate the potential energy of the block, PE_block = 0.001 kg * 9.8 m/s^2 * 0.35 m.
Step 2: Calculate the kinetic energy of the bullet.
The kinetic energy (KE) of an object is given by the equation: KE = 0.5 * mv^2, where m is the mass (in kg) and v is the velocity (in m/s).
Since the bullet embeds in the block, the bullet's kinetic energy is transferred to the block. Therefore, the kinetic energy of the bullet must equal the potential energy of the block, i.e., KE_bullet = PE_block.
Step 3: Equate the kinetic energy of the bullet with its known values and solve for velocity.
We have KE_bullet = 0.5 * mv^2, where the mass of the bullet is 5g (or 0.005 kg). Substituting the given values, we get: 0.5 * 0.005 kg * v^2 = 0.001 kg * 9.8 m/s^2 * 0.35 m.
Now we can solve this equation to find the value of v, which represents the initial velocity of the bullet before embedding in the block.
Step 4: Solve for v.
Rearrange the equation to solve for v:
0.5 * 0.005 kg * v^2 = 0.001 kg * 9.8 m/s^2 * 0.35 m
v^2 = (0.001 kg * 9.8 m/s^2 * 0.35 m) / (0.5 * 0.005 kg)
v^2 = 0.196 m^2/s^2
v = √(0.196) m/s
Using a calculator, compute the square root of 0.196:
v ≈ 0.442 m/s
Therefore, the bullet was originally moving at a speed of approximately 0.442 m/s.
Remember to always check your units and make sure they are consistent throughout the calculations.