A 22.0-kg body is moving in the direction of the positive x axis with a speed of 359 m/s when, owing to an internal explosion, it breaks into three pieces. One part, whose mass is 6.5 kg, moves away from the point of explosion with a speed of 387 m/s along the positive y axis. A second fragment, whose mass is 3.5 kg, moves away from the point of explosion with a speed of 458 m/s along the negative x axis. What is the speed of the third fragment? Ignore effects due to gravity.
Well, it sounds like the explosion really blew the body into pieces. Talk about a splitting headache! Anyway, let's solve this math puzzle!
To find the speed of the third fragment, we need to use the law of conservation of momentum. The total momentum before the explosion must be equal to the total momentum after the explosion.
Before the explosion, the total momentum is given by the mass of the body (22.0 kg) multiplied by its initial velocity (359 m/s). So, we have:
Total momentum before = 22.0 kg * 359 m/s
After the explosion, the total momentum is the sum of the individual momenta of the three fragments. Let's call the mass of the third fragment "m3" and its speed "v3". The momentum of the first fragment is given by the mass (6.5 kg) times its speed (387 m/s) in the positive y direction, while the momentum of the second fragment is given by the mass (3.5 kg) times its speed (458 m/s) in the negative x direction.
Total momentum after = (6.5 kg * 387 m/s) + (3.5 kg * -458 m/s) + (m3 * v3)
According to the conservation of momentum, the total momentum before the explosion must be equal to the total momentum after the explosion. So, we have:
22.0 kg * 359 m/s = (6.5 kg * 387 m/s) + (3.5 kg * -458 m/s) + (m3 * v3)
Now, let's solve for v3:
(22.0 kg * 359 m/s) - (6.5 kg * 387 m/s) + (3.5 kg * 458 m/s) = m3 * v3
Plugging in the given values and solving the equation will give us the speed of the third fragment. Good luck, and remember to keep your pieces together!
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the explosion is equal to the sum of the momenta of the three fragments after the explosion. Mathematically, this can be written as:
m1v1 + m2v2 + m3v3 = m'1v'1 + m'2v'2 + m'3v'3
where m1, m2, and m3 are the masses of the three fragments before the explosion, v1, v2, and v3 are their respective velocities before the explosion, and m'1, m'2, and m'3 are their masses after the explosion, and v'1, v'2, and v'3 are their respective velocities after the explosion.
In this case, we are given:
m1 = 22.0 kg
v1 = 359 m/s
m'1 = 6.5 kg
v'1 = 387 m/s
m'2 = 3.5 kg
v'2 = -458 m/s
We need to find v'3.
Let's substitute these values into the conservation of momentum equation and solve for v'3:
(22.0 kg)(359 m/s) + 0 + 0 = (6.5 kg)(387 m/s) + (3.5 kg)(-458 m/s) + (m'3)(v'3)
After simplifying:
7968 kg*m/s = 2514.5 kg*m/s - 1603 kg*m/s + (m'3)(v'3)
Rearranging the equation:
1603 kg*m/s + 2514.5 kg*m/s = (m'3)(v'3)
4117.5 kg*m/s = (m'3)(v'3)
Now, we can calculate the velocity of the third fragment, v'3, by dividing both sides of the equation by m'3:
v'3 = (4117.5 kg*m/s) / m'3
Since we are not given the value of m'3, we cannot calculate the exact velocity of the third fragment. However, we can provide a general equation to find v'3:
v'3 = 4117.5 / m'3
So, the speed of the third fragment, v'3, depends on the mass of that fragment, m'3.
To find the speed of the third fragment, we can use the principle of conservation of momentum. This principle states that the total momentum before the explosion is equal to the total momentum after the explosion.
Considering a coordinate system where the positive x-axis is to the right and the positive y-axis is upwards, we can break down the initial and final momenta as follows:
Initial momentum (before the explosion):
P_initial = P_body = m_body * v_body
Final momentum (after the explosion):
P_final = P_fragment_1 + P_fragment_2 + P_fragment_3
Here, P_fragment_1, P_fragment_2, and P_fragment_3 represent the momenta of the three fragments after the explosion.
We are given the masses and velocities of fragments 1 and 2, so we can calculate their momenta:
P_fragment_1 = m_fragment_1 * v_fragment_1
P_fragment_2 = m_fragment_2 * v_fragment_2
To find the speed of the third fragment, we need to determine its momentum.
First, let's calculate the magnitude of the momentum of the body before the explosion:
P_body = m_body * v_body
= 22.0 kg * 359 m/s
= 7918 Ns
Now, let's calculate the magnitude of the total momentum after the explosion:
P_final = P_fragment_1 + P_fragment_2 + P_fragment_3
Since we know the masses and velocities of fragments 1 and 2, we can calculate their momenta:
P_fragment_1 = m_fragment_1 * v_fragment_1
= 6.5 kg * 387 m/s
≈ 2515.5 Ns
P_fragment_2 = m_fragment_2 * v_fragment_2
= 3.5 kg * 458 m/s
≈ 1603 Ns
Now, we can substitute these values into the equation for P_final and solve for P_fragment_3:
P_final = P_fragment_1 + P_fragment_2 + P_fragment_3
7918 Ns = 2515.5 Ns + 1603 Ns + P_fragment_3
Rearranging the equation, we find:
P_fragment_3 = 7918 Ns - 2515.5 Ns - 1603 Ns
= 3800.5 Ns
Finally, we can find the speed of the third fragment by dividing its momentum by its mass:
P_fragment_3 = m_fragment_3 * v_fragment_3
3800.5 Ns = m_fragment_3 * v_fragment_3
Dividing both sides of the equation by m_fragment_3:
v_fragment_3 = 3800.5 Ns / m_fragment_3
We are not given the mass of the third fragment, so we cannot determine the exact speed without this information.