Two carts with masses of 11.7 kg and 2.5 kg

move in opposite directions on a frictionless
horizontal track with speeds of 6.6 m/s and
3.1 m/s, respectively. The carts stick together
after colliding head-on.
Find their final speed.
Answer in units of m/s

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Learn to apply the law of conservation of momentum. It is all you need to solve this problem.

To find the final speed of the carts after they stick together, we can apply the law of conservation of momentum. According to the principle of momentum conservation, the total momentum before the collision should be equal to the total momentum after the collision.

The formula for momentum is given by:

momentum = mass * velocity

Let's calculate the momentum of each cart before the collision:

Momentum of the first cart (m1) = mass of the first cart * velocity of the first cart
= 11.7 kg * 6.6 m/s

Momentum of the second cart (m2) = mass of the second cart * velocity of the second cart
= 2.5 kg * 3.1 m/s

The total momentum before the collision is the sum of the individual momenta:

Initial total momentum = m1 + m2

Now, after the collision, the two carts stick together and move with a final velocity (vf). Therefore, the total mass of the system is the sum of the individual masses:

Total mass (M) = m1 + m2

According to the principle of conservation of momentum, the initial total momentum should be equal to the final total momentum:

Initial total momentum = Final total momentum
Initial total momentum = m1 + m2 = M * vf

Since the initial total momentum is known, we can rearrange the equation to find the final velocity (vf):

vf = (m1 + m2) / M

Substituting the given values:

vf = (11.7 kg * 6.6 m/s + 2.5 kg * 3.1 m/s) / (11.7 kg + 2.5 kg)

Simplifying the expression:

vf = (77.22 kg·m/s + 7.75 kg·m/s) / 14.2 kg

vf = 84.97 kg·m/s / 14.2 kg

vf ≈ 5.99 m/s (rounded to two decimal places)

Therefore, the final speed of the carts, when they stick together after the head-on collision, is approximately 5.99 m/s.