An 7.6-kg object is sliding across the ice at 2.34 m/s. An internal explosion occurs, splitting the object into two equal chunks and adding 11 J of kinetic energy to the system. What is the average acceleration of the two chunks if the explosive separation takes place over a 0.16-s time interval?
To find the average acceleration of the two chunks, we first need to find the final velocity of each chunk after the explosion.
Let's assume that the mass of each chunk is m kg.
Since the initial object is sliding across the ice without any external forces acting on it, we can consider the conservation of momentum before and after the explosion. The initial momentum of the object is given by:
Initial momentum = mass × initial velocity
= (7.6 kg) × (2.34 m/s)
After the explosion, the final momentum of the system is given by:
Final momentum = 2 × mass × final velocity
= 2 × m kg × vf
According to the law of conservation of momentum, the initial momentum must be equal to the final momentum. Therefore:
(7.6 kg) × (2.34 m/s) = 2 × m kg × vf
Simplifying the equation:
16.824 kg·m/s = 2mvf
Now, let's calculate the final velocity, vf, by using the given information that 11 J of kinetic energy is added to the system. We can relate the change in kinetic energy with the final velocity using the equation:
ΔKE = (1/2) × mass × (vf^2 - vi^2)
Substituting the given values:
11 J = (1/2) × (2m kg) × (vf^2 - 0^2)
11 J = m kg × vf^2
Simplifying the equation:
11 J = 2mvf^2
Now we can combine the equations for final momentum and the final velocity:
16.824 kg·m/s = 2mvf
11 J = 2mvf^2
Since the mass of each chunk is the same, we can equate the expressions for vf:
16.824 kg·m/s = 11 J^(1/2) × vf
To find vf, we can rearrange the equation:
vf = (16.824 kg·m/s) / (11 J^(1/2))
Now, we can find the average acceleration, which is the change in velocity divided by the time interval. The change in velocity is the final velocity minus the initial velocity:
Change in velocity = vf - vi
= vf - 2.34 m/s
The average acceleration can be calculated as:
Average acceleration = (Change in velocity) / (Time interval)
= (vf - 2.34 m/s) / (0.16 s)