1) Blood is pumped from the heart at a rate
of 5.0 L/min into the aorta (of radius 1.0
cm). Determine the speed of blood through
the aorta.
2)a bullet of mass 0.05kg is fired with a velocity of 200m/s into a lead block of 0.9kg. Given that the lead block can move freely, the final kinetic energy after impact is?.
1) (5000 cm^3 / min) / (aorta cross sectional area)
2) momentum is conserved
(velocity B+b) * (mass B+b) = (velocity b) * (mass b)
final K.E. ... 1/2 (mass B+b) * (velocity B+b)^2
1) To determine the speed of blood through the aorta, we can use the equation of continuity, which states that the volume flow rate of an incompressible fluid is constant.
The volume flow rate (Q) can be calculated using the equation:
Q = A * v
where:
Q = Volume flow rate
A = Cross-sectional area of the aorta
v = Velocity of blood through the aorta
Since the radius (r) of the aorta is given as 1.0 cm, we can calculate the cross-sectional area (A) using the formula:
A = π * r^2
A = π * (0.01 m)^2
= 0.000314 m^2
Now, we can rearrange the equation of continuity to solve for the velocity (v):
v = Q / A
Given that the volume flow rate (Q) is 5.0 L/min, we need to convert it to m^3/s:
Q = 5.0 L/min
= 5.0 × 10^(-3) m^3/s
Now substitute the values into the equation:
v = (5.0 × 10^(-3) m^3/s) / (0.000314 m^2)
≈ 15.92 m/s
Therefore, the speed of blood through the aorta is approximately 15.92 m/s.
2) To find the final kinetic energy after impact, we need to apply the law of conservation of momentum.
The initial momentum before impact is given by the equation:
p initial = m bullet * v bullet
where:
m bullet = mass of the bullet
v bullet = velocity of the bullet
Given that the mass of the bullet (m bullet) is 0.05 kg and the velocity (v bullet) is 200 m/s:
p initial = (0.05 kg) * (200 m/s)
= 10 kg·m/s
The final momentum after impact is equal to the initial momentum, since momentum is conserved:
p final = p initial
= 10 kg·m/s
Now, the final kinetic energy after impact can be calculated using the equation:
KE final = (1/2) * m block * v block^2
where:
m block = mass of the lead block
v block = final velocity of the lead block
Given that the mass of the lead block (m block) is 0.9 kg, we need to find the final velocity (v block) of the lead block.
Since momentum is conserved, we can write:
p final = m bullet * v bullet + m block * v block
Substituting the known values:
10 kg·m/s = (0.05 kg) * (200 m/s) + (0.9 kg) * v block
Simplifying the equation for v block:
v block = (10 kg·m/s - (0.05 kg) * (200 m/s)) / (0.9 kg)
≈ 10.06 m/s
Now substitute the values into the equation for final kinetic energy:
KE final = (1/2) * (0.9 kg) * (10.06 m/s)^2
≈ 45.39 J
Therefore, the final kinetic energy after impact is approximately 45.39 Joules.
To determine the speed of blood through the aorta, we can use the equation for the volume flow rate of a fluid, which is given by:
Q = A * v
Where:
Q is the volume flow rate
A is the cross-sectional area of the aorta
v is the speed of blood through the aorta
First, let's calculate the cross-sectional area of the aorta:
A = π * r^2
Where:
π is a mathematical constant approximately equal to 3.14159
r is the radius of the aorta
In this case, the radius of the aorta is given as 1.0 cm, but we need to convert it to meters:
r = 1.0 cm = 0.01 m
Now we can calculate the cross-sectional area:
A = π * (0.01 m)^2
Next, let's substitute the given value for the volume flow rate, which is 5.0 L/min. However, we need to convert it to liters per second:
Q = 5.0 L/min = (5.0 L/min) * (1 min / 60 s) = 5.0/60 L/s
Now we can rearrange the equation to solve for the speed:
v = Q / A
Substituting the values we calculated:
v = (5.0/60 L/s) / (π * (0.01 m)^2)
Simplifying this equation will give you the speed in meters per second, which is the unit we're interested in.
For the second question, we need to find the final kinetic energy after the bullet impacts the lead block.
We can use the principle of conservation of momentum, which states that the total momentum before the impact is equal to the total momentum after the impact.
Before the impact, the bullet has a mass of 0.05 kg and a velocity of 200 m/s. The lead block has a mass of 0.9 kg and is initially at rest.
The total momentum before the impact is given by:
Total momentum before = (mass of bullet * velocity of bullet) + (mass of lead block * velocity of lead block)
After the impact, we assume the bullet and lead block move together with a final velocity v.
The total momentum after the impact is given by:
Total momentum after = (mass of bullet + mass of lead block) * final velocity
Since the total momentum before the impact is equal to the total momentum after the impact, we can set up an equation:
(mass of bullet * velocity of bullet) + (mass of lead block * velocity of lead block) = (mass of bullet + mass of lead block) * final velocity
Now, we can solve for the final velocity:
final velocity = [ (mass of bullet * velocity of bullet) + (mass of lead block * velocity of lead block) ] / (mass of bullet + mass of lead block)
Once we have the final velocity, we can calculate the final kinetic energy using the equation:
final kinetic energy = (1/2) * (mass of bullet + mass of lead block) * (final velocity)^2
Substitute the given values into the equations to find the final kinetic energy.