To solve this problem, we need to apply the principle of conservation of momentum. When the body explodes, the total momentum before the explosion must equal the total momentum after the explosion. Since the body is initially at rest, its initial momentum is zero. After the explosion, the momentum of the fragments must also sum to zero.
Understanding the Mass Distribution
The body has a total mass of 1 kg, which breaks into three fragments in the ratio of 1:1:3. Let's denote the masses of the fragments as follows:
- Fragment A: 1x
- Fragment B: 1x
- Fragment C: 3x
Since the total mass is 1 kg, we can express this as:
1x + 1x + 3x = 1 kg
This simplifies to:
5x = 1 kg
From this, we find:
x = 0.2 kg
Thus, the masses of the fragments are:
- Fragment A: 0.2 kg
- Fragment B: 0.2 kg
- Fragment C: 0.6 kg
Applying Conservation of Momentum
Since the two smaller fragments (A and B) fly off perpendicular to each other, we can analyze their momentum in two dimensions. Let’s assign directions:
- Fragment A moves along the x-axis with a speed of 15 m/s.
- Fragment B moves along the y-axis with a speed of 15 m/s.
The momentum of each fragment can be calculated as:
- Momentum of Fragment A (pA) = mass × velocity = 0.2 kg × 15 m/s = 3 kg·m/s
- Momentum of Fragment B (pB) = mass × velocity = 0.2 kg × 15 m/s = 3 kg·m/s
Calculating the Momentum of Fragment C
Since the total momentum before the explosion is zero, the momentum of Fragment C must balance the momentum of A and B. We can express the total momentum in vector form:
- Total momentum in the x-direction: pA - pC_x = 0
- Total momentum in the y-direction: pB - pC_y = 0
From the x-direction, we have:
3 kg·m/s - pC_x = 0 → pC_x = 3 kg·m/s
From the y-direction, we have:
3 kg·m/s - pC_y = 0 → pC_y = 3 kg·m/s
Finding the Speed of Fragment C
Now, we can find the speed of Fragment C using its momentum:
pC = mass × velocity
For Fragment C, we know:
pC = 0.6 kg × vC
Setting this equal to the total momentum:
3 kg·m/s = 0.6 kg × vC
Solving for vC gives:
vC = 3 kg·m/s / 0.6 kg = 5 m/s
Final Result
The speed of the heavier fragment (Fragment C) is 5 m/s.
