A 2.40 kg firecracker explodes into two pieces and produces 170 J of heat, light, and sound energy. One of the pieces has a mass of 0.800 kg and a speed of 4.50 m/s after the explosion. If the mass of the other piece is 1.20 kg, what is its speed after the explosion?
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of energy.
First, let's calculate the momentum before the explosion. The total momentum before the explosion is zero because the firecracker is initially at rest. Therefore, the total momentum after the explosion should also be zero since there is no external force acting on the system.
Let's denote the speed of the larger piece after the explosion as v1.
The momentum of the smaller piece after the explosion is given by:
p2 = m2 * v2,
where p2 is the momentum, m2 is the mass of the smaller piece, and v2 is its speed. We are given that m2 = 0.8 kg and v2 = 4.5 m/s.
Now, let's calculate the momentum of the larger piece after the explosion. The sum of the momenta of both pieces should be zero:
p1 + p2 = 0,
where p1 is the momentum of the larger piece. Therefore,
p1 = -p2.
Substituting the values:
m1 * v1 = -m2 * v2,
where m1 is the mass of the larger piece.
Next, let's consider the conservation of energy. The total energy produced during the explosion is given as 170 J. This energy is shared between the two pieces. So, we can write:
E = 0.5 * m1 * v1^2 + 0.5 * m2 * v2^2,
where E is the total energy, m1 is the mass of the larger piece, v1 is its speed, m2 is the mass of the smaller piece, and v2 is its speed.
Substituting the known values:
170 J = 0.5 * m1 * v1^2 + 0.5 * 0.8 kg * (4.5 m/s)^2.
Now we can solve these two equations simultaneously to find the speed of the larger piece (v1).
First, rearrange the momentum equation:
m1 * v1 = -(m2 * v2).
Substitute the values:
m1 * v1 = -(1.2 kg * 4.5 m/s).
Calculate the momentum:
m1 * v1 = -5.4 kg*m/s.
Now, substitute this value into the energy equation and solve for v1:
170 J = 0.5 * (-5.4 kg*m/s) * v1 + 0.5 * 0.8 kg * (4.5 m/s)^2.
Simplify the equation:
170 J = -2.7 kg*m^2/s^2 * v1 + 1.62 J.
Rearrange the equation:
2.7 kg*m^2/s^2 * v1 = 1.62 J - 170 J.
Combine the terms on the right side:
2.7 kg*m^2/s^2 * v1 = - 168.38 J.
Now, isolate v1 by dividing both sides by 2.7 kg*m^2/s^2:
v1 = (- 168.38 J) / (2.7 kg*m^2/s^2).
Calculate v1:
v1 ≈ - 62.4 m/s.
The negative sign indicates that the larger piece is moving in the opposite direction to the smaller piece. We can discard the negative sign since we are interested only in the magnitude of the speed.
Therefore, the speed of the larger piece after the explosion is approximately 62.4 m/s.
To solve this problem, we can use the principle of conservation of momentum.
1. The total momentum before the explosion is equal to the total momentum after the explosion.
2. We know that one piece has a mass of 0.800 kg and a speed of 4.50 m/s, so its momentum is given by the formula:
momentum = mass * velocity
momentum of the 0.800 kg piece = 0.800 kg * 4.50 m/s = 3.6 kg·m/s
3. The other piece has a mass of 1.20 kg. Let's assume its velocity after the explosion is "v".
4. The total momentum after the explosion is the sum of the individual momenta of the two pieces:
momentum of the 0.800 kg piece + momentum of the 1.20 kg piece = total momentum after the explosion
3.6 kg·m/s + momentum of the 1.20 kg piece = total momentum after the explosion
5. Since the total momentum after the explosion must equal the total momentum before the explosion, we can set up the equation:
total momentum before the explosion = total momentum after the explosion
0 kg·m/s + 0 kg·m/s = 3.6 kg·m/s + momentum of the 1.20 kg piece
We can simplify this equation to:
momentum of the 1.20 kg piece = -3.6 kg·m/s
The negative sign indicates that the momentum direction is opposite to the 0.800 kg piece.
6. Now we can use the momentum of the 1.20 kg piece to find its velocity.
momentum = mass * velocity
-3.6 kg·m/s = 1.20 kg * velocity
Solving for velocity, we get:
velocity = -3.6 kg·m/s / 1.20 kg
velocity ≈ -3 m/s
Note: The negative sign indicates that the direction of velocity is opposite to the 0.800 kg piece.
Therefore, the speed of the other piece, with a mass of 1.20 kg, after the explosion is approximately 3 m/s in the opposite direction.