An 9.20kg ball, hanging from the ceiling by a light wire 115 long, is struck in an elastic collision by a 1.20kg ball moving horizontally at 6.00m/s just before the collision.
Find the tension in the wire just after the collision.
92N
To find the tension in the wire just after the collision, we need to consider the conservation of momentum and the conservation of kinetic energy.
Let's start by calculating the initial momentum of each ball.
The momentum of an object is given by its mass multiplied by its velocity. The initial momentum of the 9.20 kg ball hanging from the ceiling is zero because it is at rest.
The initial momentum of the 1.20 kg ball moving horizontally at 6.00 m/s can be calculated as follows:
Initial momentum = mass × velocity
Initial momentum = 1.20 kg × 6.00 m/s
Initial momentum = 7.20 kg·m/s
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. After the collision, the 9.20 kg ball and the 1.20 kg ball will move together as one system.
Next, we need to consider the conservation of kinetic energy. In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The kinetic energy of an object is given by one-half times its mass times the square of its velocity. The initial kinetic energy of the 1.20 kg ball is:
Initial kinetic energy = 0.5 × mass × (velocity)^2
Initial kinetic energy = 0.5 × 1.20 kg × (6.00 m/s)^2
Initial kinetic energy = 21.6 J
Since the 9.20 kg ball is initially at rest, its initial kinetic energy is zero.
After the collision, the 9.20 kg ball and the 1.20 kg ball will be moving together with a common velocity. We can denote this common velocity as V.
Now, we can equate the total momentum before the collision to the total momentum after the collision and solve for V.
Total initial momentum = Total final momentum
0 + 7.20 kg·m/s = (9.20 kg + 1.20 kg) × V
7.20 kg·m/s = 10.40 kg × V
V = 7.20 kg·m/s / 10.40 kg
V ≈ 0.692 m/s
Now that we know the common velocity after the collision, we can find the tension in the wire. The tension in the wire provides the force necessary to accelerate the system (9.20 kg + 1.20 kg) with a common velocity.
The tension force in the wire can be calculated using Newton's second law, which states that force is equal to mass times acceleration:
Tension force = (9.20 kg + 1.20 kg) × acceleration
Acceleration, in this case, is the change in velocity divided by the time taken to achieve that change. Since the 9.20 kg + 1.20 kg system starts from rest and achieves a velocity of 0.692 m/s, we can use the final velocity and divide it by the time it takes to reach that velocity.
The time taken by the system to reach the final velocity can be calculated using the equation for average acceleration:
final velocity = initial velocity + (acceleration × time)
Since the initial velocity is 0 m/s and the final velocity is 0.692 m/s, we have:
0.692 m/s = 0 m/s + (acceleration × time)
Now, we need to solve for time.
time = (0.692 m/s - 0 m/s) / acceleration
To calculate acceleration, we can use the equation:
acceleration = change in velocity / time
Since the change in velocity is 0.692 m/s (final velocity) and the time is the value we are solving for, we can substitute these values into the equation to obtain:
acceleration = 0.692 m/s / time
We can now substitute the acceleration value in terms of time back into the equation for time to solve for it:
time = (0.692 m/s - 0 m/s) / (0.692 m/s / time)
time = 1 s
Therefore, it takes 1 second for the system to reach the final velocity of 0.692 m/s.
Finally, we can calculate the tension force in the wire using Newton's second law:
Tension force = (9.20 kg + 1.20 kg) × acceleration
Tension force = 10.40 kg × (0.692 m/s / 1 s)
Tension force ≈ 6.80 N
So, the tension in the wire just after the collision is approximately 6.80 N.