A 5.00 kg ball (ball 1), moving to the right at a velocity of +4.00 m/s on a frictionless table, collides head-on with a stationary 7.60 kg ball (ball 2). Find the final velocities of the balls if the collision is as specified below.
(a) elastic collision
ball 1
ball 2
(b) completely inelastic collision
ball 1
ball 2
i solved part a but cant figure out part b. if i make the assumption that in a completley ineleastic equation the balls are stuck togtether and have the same velocity, how would i go about solving it.
In a completely inelastic collision, the two colliding objects stick together and move as a single unit after the collision. To solve for the final velocities of the balls in a completely inelastic collision, you need to apply the principle of conservation of momentum.
Step 1: Determine the initial momentum of the system before the collision.
The momentum (p) of an object is defined as the product of its mass (m) and velocity (v), so the initial momentum of ball 1 is:
p1(initial) = m1 * v1
= 5.00 kg * 4.00 m/s
= 20.00 kg·m/s
The initial momentum of ball 2 is zero since it is initially stationary:
p2(initial) = m2 * v2
= 7.60 kg * 0 m/s
= 0 kg·m/s
The total initial momentum of the system is the sum of the individual momenta:
p(initial) = p1(initial) + p2(initial)
= 20.00 kg·m/s + 0 kg·m/s
= 20.00 kg·m/s
Step 2: Determine the final momentum of the system after the collision.
In a completely inelastic collision, the colliding objects stick together, so they have a common final velocity (vf).
The final momentum of the system is given by:
p(final) = (m1 + m2) * vf
Step 3: Apply the conservation of momentum principle.
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
So, we can equate the initial momentum to the final momentum:
p(initial) = p(final)
Substituting the values,
20.00 kg·m/s = (5.00 kg + 7.60 kg) * vf
Simplifying the equation,
20.00 kg·m/s = 12.60 kg * vf
Dividing both sides by 12.60 kg,
vf = 20.00 kg·m/s / 12.60 kg
= 1.58 m/s
Therefore, in a completely inelastic collision:
ball 1 and ball 2 will have a final velocity of 1.58 m/s
To solve part (b) of the problem, which is a completely inelastic collision, you correctly assume that the two balls stick together and move as a single object with the same final velocity.
In a completely inelastic collision, momentum is conserved, just like in an elastic collision. However, in an inelastic collision, kinetic energy is not conserved and is lost as heat, sound, or deformation of the objects.
To solve the problem, you can follow these steps:
1. Write down the initial momentum of the system before the collision:
Initial momentum = (mass of ball 1) * (velocity of ball 1) + (mass of ball 2) * (velocity of ball 2)
= (5.00 kg) * (+4.00 m/s) + (7.60 kg) * (0 m/s)
= 20.00 kg·m/s + 0 kg·m/s
= 20.00 kg·m/s
2. After the collision, since the balls stick together, they have the same final velocity. Let's call this final velocity 'Vf'.
3. Write down the final momentum of the system after the collision:
Final momentum = (mass of the combined balls) * (final velocity)
= (5.00 kg + 7.60 kg) * Vf
= 12.60 kg * Vf
4. Apply the principle of conservation of momentum:
Initial momentum = Final momentum
20.00 kg·m/s = 12.60 kg * Vf
5. Solve for Vf:
Vf = 20.00 kg·m/s / 12.60 kg
≈ 1.59 m/s
Therefore, in a completely inelastic collision, both ball 1 and ball 2 stick together and move with a final velocity of approximately 1.59 m/s.