Two carts with masses of 4.24 kg and 2.22 kg move toward each other on a frictionless track with speeds of 4.55 m/s and 3.16 m/s respectively. The carts stick together after colliding head-on. Find the final speed.
4.24 * 4.55 - 2.22 * 3.16 = (4.24+2.22) v
Thank you(((:
To find the final speed of the carts after the collision, we can apply the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant before and after a collision. In this case, the carts collide and stick together, forming a single system.
The momentum of an object is given by the product of its mass and velocity. Mathematically, the momentum (p) is equal to mass (m) multiplied by velocity (v):
p = m * v
Before the collision, the momentum of the first cart is:
p1 = m1 * v1 = 4.24 kg * 4.55 m/s
Before the collision, the momentum of the second cart is:
p2 = m2 * v2 = 2.22 kg * (-3.16 m/s) (negative sign as it is moving in the opposite direction)
Since momentum is conserved, the total momentum of the system after the collision will be equal to the sum of the momenta of the individual carts before the collision:
p_total = p1 + p2
Now, we can calculate the total momentum and find the final velocity (v_final) of the combined carts:
p_total = (m1 + m2) * v_final
Setting the two expressions for p_total equal to each other, we have:
(m1 + m2) * v_final = p1 + p2
Simplifying, we can solve for v_final:
v_final = (p1 + p2) / (m1 + m2)
Plugging in the given values, we get:
v_final = (4.24 kg * 4.55 m/s + 2.22 kg * (-3.16 m/s)) / (4.24 kg + 2.22 kg)
Calculating the numerator:
4.24 kg * 4.55 m/s + 2.22 kg * (-3.16 m/s) = (19.2920 kg*m/s + (-7.0032 kg*m/s))
Calculating the denominator:
4.24 kg + 2.22 kg = 6.46 kg
Adding the numerator and dividing by the denominator, we find:
v_final = (19.2920 kg*m/s + (-7.0032 kg*m/s)) / 6.46 kg
v_final = 12.2888 kg*m/s / 6.46 kg
v_final ≈ 1.90 m/s
Therefore, the final speed of the combined carts after the collision is approximately 1.90 m/s.