An object at rest explodes into three fragments. The figure shows the momentum vectors of two of the fragments:
vector P1 is directed along the px axis and has a magnitude of 74.0 kg m/s.
vector P2 is at an angle of 300.° measured counterclockwise from the positive px axis and has a magnitude of 26.0 kg m/s.
What are the px and py components of the momentum of the third fragment?
To find the px and py components of the momentum of the third fragment, we will use vector addition.
Given:
P1 = 74.0 kg m/s along the px axis
P2 = 26.0 kg m/s at an angle of 300° counterclockwise from the positive px axis
First, let's find the px and py components of P2:
px component of P2 = P2 * cos(300°)
py component of P2 = P2 * sin(300°)
px component of P2 = 26.0 kg m/s * cos(300°)
px component of P2 = 26.0 kg m/s * (-0.5)
px component of P2 = -13.0 kg m/s
py component of P2 = 26.0 kg m/s * sin(300°)
py component of P2 = 26.0 kg m/s * (-√3/2)
py component of P2 = -22.5 kg m/s
Next, let's find the px and py components of the third fragment by adding the px and py components of P1 and P2:
px component of third fragment = px component of P1 + px component of P2
py component of third fragment = py component of P1 + py component of P2
px component of third fragment = 74.0 kg m/s + (-13.0 kg m/s)
px component of third fragment = 61.0 kg m/s
py component of third fragment = 0 kg m/s + (-22.5 kg m/s)
py component of third fragment = -22.5 kg m/s
Therefore, the px component of the momentum of the third fragment is 61.0 kg m/s, and the py component is -22.5 kg m/s.
To find the px and py components of the momentum of the third fragment, we can start by analyzing the given information.
We know that the total momentum before the explosion is equal to the total momentum after the explosion. In other words, the vector sum of the momenta of the fragments before the explosion is equal to the vector sum of the momenta of the fragments after the explosion.
Let's assume that the px and py components of the momentum of the third fragment are P3x and P3y respectively.
The momentum of the first fragment, P1, is directed along the px axis, so its py component is zero. Therefore, the px and py components of P1 are 74.0 kg m/s and 0 kg m/s respectively.
The momentum of the second fragment, P2, is at an angle of 300° counterclockwise from the positive px axis. To find its px and py components, we can use trigonometry.
The px component of P2 (P2x) can be found using the cosine function:
P2x = P2 * cos(300°)
Substituting the given values:
P2x = 26.0 kg m/s * cos(300°)
Now, the py component of P2 (P2y) can be found using the sine function:
P2y = P2 * sin(300°)
Substituting the given values:
P2y = 26.0 kg m/s * sin(300°)
Finally, the total momentum after the explosion is the vector sum of the momenta of the three fragments. Therefore, the px and py components of the total momentum after the explosion are:
Px_total = P1x + P2x + P3x
Py_total = P1y + P2y + P3y
Since we know the values of Px_total, P1x, P2x, Py_total, P1y, and P2y, we can rearrange these equations to find P3x and P3y:
P3x = Px_total - P1x - P2x
P3y = Py_total - P1y - P2y
By substituting the given values and solving these equations, we can find the px and py components of the momentum of the third fragment.
If it was at rest and there is no EXTERNAL force then:
total x momentum = 0
total y momentum = 0