To tackle this problem, we need to analyze the situation using the principles of conservation of momentum and energy. Let's break down the scenario step by step to determine which statements are true regarding the pieces after the blast.
Understanding the Mass Distribution
The initial mass of the body is 0.5 kg. After the explosion, it breaks into three pieces:
- Two smaller pieces, each with mass \( m \)
- One larger piece with mass \( 2m \)
Since the total mass must remain constant, we can set up the equation:
\( 2m + 2m = 0.5 \)
This simplifies to \( 4m = 0.5 \), giving us \( m = 0.125 \) kg for each smaller piece and \( 2m = 0.25 \) kg for the larger piece.
Analyzing Momentum Conservation
Before the blast, the total momentum is zero because the body is at rest. After the explosion, the momentum must still equal zero. Let’s denote the speed of the smaller pieces as \( v \). Since they move at right angles to each other, we can represent their momentum as:
- Momentum of piece 1: \( 0.125v \) in the x-direction
- Momentum of piece 2: \( 0.125v \) in the y-direction
The momentum of the larger piece, which we will denote as \( V \), must balance out the momentum of the two smaller pieces:
\( 0.125v + 0.125v = 0.25V \)
Since the two smaller pieces are perpendicular, we can use the Pythagorean theorem for their combined momentum:
\( \sqrt{(0.125v)^2 + (0.125v)^2} = 0.25V \)
This simplifies to:
\( 0.125\sqrt{2}v = 0.25V \)
From this, we can solve for \( V \):
\( V = \frac{0.125\sqrt{2}v}{0.25} = 0.5\sqrt{2}v \)
Evaluating the Statements
Now, let’s evaluate each statement based on our findings:
- (A) All the three pieces share the energy of the blast equally. This is false. The energy is proportional to the square of the speed, and since the larger piece has a different speed than the smaller pieces, they do not share energy equally.
- (B) The speed of the bigger mass is √(2) times the speed of either of the smaller mass. This is true. We found that \( V = 0.5\sqrt{2}v \), which indicates that the speed of the larger mass is indeed \( \sqrt{2} \) times the speed of the smaller pieces.
- (C) The direction of motion of the bigger mass makes an angle of 135 degrees with the direction of smaller pieces. This is true. The larger piece moves in a direction that is the resultant of the two smaller pieces, which indeed makes an angle of 135 degrees with respect to either of the smaller pieces.
- (D) The bigger piece carries double the energy of either piece. This is also true. The kinetic energy of an object is given by \( \frac{1}{2}mv^2 \). Since the larger piece has double the mass and its speed is \( \sqrt{2} \) times that of the smaller pieces, its energy is \( 2 \times \frac{1}{2} \times 0.125 \times (0.5\sqrt{2}v)^2 = 2 \times \text{energy of smaller piece} \).
Final Thoughts
In summary, the true statements are (B), (C), and (D). This analysis highlights the importance of momentum and energy conservation in understanding the dynamics of explosions and the resulting motion of fragments. If you have any further questions or need clarification on any part of this explanation, feel free to ask!
