A ball of mass m at rest at the coordinate origin exploded into three equal pieces. At a certain instant, one piece is on the x-axis at x=40cm, and another is at x=20cm, y= -60 cm. Where is the third piece at that instant.
The initial momentum is zero, so the final vector momentum is zero. You have the positions, dividing by time t means velocity.
in the x:
0=1/3 40/t + 1/3 20/t + 1/3 x/t
multipy both sides by t, solve for x.
in the y:
0=1/3 0 + 1/3 (-60/t) + 1/3 y/t
solve for y.
you have x,y coordinate of the third piece at that time t.
IS the answer going to be
(-60,60)
To find the position of the third piece, we can use the conservation of momentum. If the ball was initially at rest, the total momentum before and after the explosion should be the same.
Let's assume that the x-axis is positive towards the right and the y-axis is positive upwards.
Given:
Mass of the ball, m
Position of first piece, P1: x = 40 cm (x-axis), y = 0 cm (y-axis)
Position of second piece, P2: x = 20 cm (x-axis), y = -60 cm (y-axis)
Since the ball exploded into three equal pieces, each piece has a mass of m/3.
Let the position of the third piece, P3, be (x3, y3).
To find the position of the third piece, we need to determine its coordinates separately in the x-direction and y-direction.
In the x-direction, we can use the conservation of momentum:
Initial momentum = Final momentum
0 = (m/3) * 0 + (m/3) * 40 + (m/3) * x3
Simplifying the equation:
0 = 40 + x3
x3 = -40 cm
Therefore, the x-coordinate of the third piece, P3, is -40 cm.
Now, let's find the y-coordinate of the third piece, P3.
In the y-direction, we can again use the conservation of momentum:
Initial momentum = Final momentum
0 = (m/3) * 0 + (m/3) * 0 + (m/3) * y3
Simplifying the equation:
0 = y3
y3 = 0 cm
Therefore, the y-coordinate of the third piece, P3, is 0 cm.
Hence, the third piece is located at (-40 cm, 0 cm) at that instant.
To find the location of the third piece, we can make use of the conservation of momentum. Since the initial ball was at rest, the total momentum before the explosion is zero. After the explosion, the momentum of the three pieces should add up to zero as well.
Let's denote the mass of each piece as m1, m2, and m3. According to the problem, all three pieces are equal, so we have m1 = m2 = m3 = m.
Now, let's assign coordinates to each piece. The first piece is located on the x-axis at x = 40 cm, which means its y-coordinate is 0. The second piece is located at x = 20 cm and y = -60 cm.
Let's denote the x and y coordinates of the third piece as x3 and y3, respectively. Since the total momentum after the explosion is zero, we can write the equation:
m1 * x1 + m2 * x2 + m3 * x3 = 0
In this case, we have:
m * 40 cm + m * 20 cm + m3 * x3 = 0
Simplifying the equation gives:
60 m * cm + m3 * x3 = 0
Since all three pieces have equal mass, we can substitute m3 with m:
60 m * cm + m * x3 = 0
To find x3, we rearrange the equation:
x3 = -60 cm
So, the third piece is located at x = -60 cm.
Therefore, the third piece is at x = -60 cm and y = 0 cm.