Two carts with masses of 4.40 kg and 2.68 kg move toward each other on a frictionless track with speeds of 5.14 m/s and 3.04 m/s respectively. The carts stick together after colliding head-on. Find the final speed.
Question
Two carts with masses of 4.40 kg and 2.68 kg move toward each other on a frictionless track with speeds of 5.14 m/s and 3.04 m/s respectively. The carts stick together after colliding head-on. Find the final speed.
Solution
Sure, let's solve this step by step.
This is a problem of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision because there are no external forces.
The momentum of an object is given by the product of its mass and its velocity (p = mv).
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Calculate the initial momentum of each cart:
For the 4.40 kg cart: p1_initial = m1 * v1 = 4.40 kg * 5.14 m/s = 22.616 kg*m/s
For the 2.68 kg cart: p2_initial = m2 * v2 = 2.68 kg * (-3.04 m/s) = -8.1472 kg*m/s
(We take the velocity of the second cart as negative because it is moving in the opposite direction)
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Calculate the total initial momentum:
p_total_initial = p1_initial + p2_initial = 22.616 kgm/s - 8.1472 kgm/s = 14.4688 kg*m/s
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After the collision, the carts stick together and move with a common velocity (v_final). The total final momentum is:
p_total_final = (m1 + m2) * v_final
But from the conservation of momentum, we know that p_total_initial = p_total_final. Therefore, we can write:
(m1 + m2) * v_final = p_total_initial
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Solve the above equation for v_final:
v_final = p_total_initial / (m1 + m2) = 14.4688 kg*m/s / (4.40 kg + 2.68 kg) = 2.24 m/s
So, the final speed of the carts after the collision is 2.24 m/s.
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