A 0.03-kg bullet is fired vertically at 200m/s into a 0.15-kg baseball that is initially at rest. The bullet lodges in the baseball and, after the collision, the baseball/bullet rise to a height of 37m. (a). What was the speed of the baseball/bullet right after the collision? (b). What was the average force of air resistance while the baseball/bullet was rising?
To solve this problem, we can apply the principles of conservation of momentum and conservation of energy. Let's break down the steps:
Step 1: Calculate the initial momentum of the bullet:
The initial momentum of the bullet can be calculated using the formula:
p_initial = m_bullet * v_bullet
Where:
m_bullet = mass of the bullet = 0.03 kg
v_bullet = velocity of the bullet = 200 m/s
So, the initial momentum of the bullet is:
p_initial = 0.03 kg * 200 m/s = 6 kg*m/s
Step 2: Calculate the initial momentum of the baseball:
Since the baseball is initially at rest, its initial momentum is zero.
p_initial_baseball = 0 kg*m/s
Step 3: Calculate the total initial momentum:
The total initial momentum before the collision is the sum of the bullet's and baseball's initial momentum.
p_initial_total = p_initial_bullet + p_initial_baseball
= 6 kg*m/s + 0 kg*m/s
= 6 kg*m/s
Step 4: Calculate the final momentum:
The final momentum can be calculated using the equation for conservation of momentum:
p_final_total = p_initial_total
Where the total momentum after the collision is equal to the total momentum before the collision.
p_final_total = m_bullet * v_final_bullet + m_baseball * v_final_baseball
Since the bullet lodges in the baseball, the final velocity of the bullet and the baseball are the same.
p_final_total = (m_bullet + m_baseball) * v_final
Step 5: Solve for the final velocity (v_final):
Rearranging the equation from step 4, we get:
v_final = p_final_total / (m_bullet + m_baseball)
Substituting the given values, we have:
v_final = (6 kg*m/s) / (0.03 kg + 0.15 kg)
= (6 kg*m/s) / (0.18 kg)
= 33.3 m/s
Therefore, the speed of the baseball/bullet right after the collision is 33.3 m/s.
Step 6: Calculate the change in potential energy:
The change in potential energy can be calculated using the formula:
∆PE = m * g * ∆h
Where:
m = total mass of the system = m_bullet + m_baseball = 0.03 kg + 0.15 kg = 0.18 kg
g = acceleration due to gravity = 9.8 m/s²
∆h = change in height = 37 m
∆PE = (0.18 kg) * (9.8 m/s²) * (37 m)
= 63.756 J
Step 7: Calculate the work done by air resistance:
The work done by air resistance is equal to the change in potential energy:
Work = ∆PE = 63.756 J
Step 8: Calculate the average force of air resistance:
The average force of air resistance can be calculated using the formula:
Force = Work / Distance
In this case, the distance is equal to the change in height (37 m).
Force = 63.756 J / 37 m
= 1.721 N
Therefore, the average force of air resistance while the baseball/bullet was rising is 1.721 N.
To find the speed of the baseball/bullet right after the collision, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
(a). Step-by-step solution to find the speed of the baseball/bullet right after the collision:
1. Write down the given information:
Mass of the bullet (m1) = 0.03 kg
Initial velocity of the bullet (v1) = 200 m/s
Mass of the baseball (m2) = 0.15 kg
Initial velocity of the baseball (v2) = 0 m/s
2. Calculate the initial momentum:
Initial momentum = (Mass of the bullet × Initial velocity of the bullet) + (Mass of the baseball × Initial velocity of the baseball)
Momentum before the collision = (0.03 kg × 200 m/s) + (0.15 kg × 0 m/s)
3. Calculate the final momentum:
Since the bullet lodges in the baseball and moves together, the final momentum is equal to the momentum of the combined baseball and bullet system.
Final momentum = (Mass of the baseball/bullet × Final velocity of the baseball/bullet)
4. Apply the principle of conservation of momentum:
Initial momentum = Final momentum
(0.03 kg × 200 m/s) + (0.15 kg × 0 m/s) = (Total mass of the baseball/bullet × Final velocity of the baseball/bullet)
5. Solve for the final velocity of the baseball/bullet:
Final velocity of the baseball/bullet = (0.03 kg × 200 m/s) / (0.03 kg + 0.15 kg)
6. Calculate the final velocity of the baseball/bullet.
Next, to find the average force of air resistance while the baseball/bullet was rising, we can use the work-energy principle. The work done by the force of air resistance is equal to the change in kinetic energy.
(b). Step-by-step solution to find the average force of air resistance:
1. Write down the given information:
Mass of the baseball and bullet system (m) = mass of the bullet + mass of the baseball
Change in height (h) = 37 m
2. Calculate the change in gravitational potential energy:
Change in potential energy = (Mass of the baseball/bullet system × g × Change in height)
Change in potential energy = (m × 9.8 m/s^2 × h)
3. Use the work-energy principle:
Work done by air resistance = Change in kinetic energy
Work done by air resistance = (1/2 × m × (Final velocity of the baseball/bullet)^2) - (1/2 × m × (Initial velocity of the baseball/bullet)^2)
4. Solve for the average force of air resistance:
Average force of air resistance = Work done by air resistance / Change in distance (which is equal to the change in height)
5. Calculate the average force of air resistance.
(a) The actual speed V' of the bullet/ball combination after impact can be obtained by applying conservation of momentum.
m*200 = (m+M) V'
V' = (0.03/0.018)*200 = 33.3 m/s
If there were no air resistance, the ball and bullet would rise to a height H given by
(m+M)gH = (1/2)(m+M)V'^2
H = V'^2/(2g) = 56.7 m
Since it only rose 37 m, 19.7/56.7 = 34.7% of the ball/bullet's initial kinetic energy is lost due to air friction. That would be
(1/2)(m+M)*V'^2*(0.347)
= 34.6 Joules
Set that energy loss equal to Fav*37 m to get the average air friction force, Fav = 0.936 Newtons