Two carts with masses of 4.66 kg and 3.00 kg move toward each other on a frictionless track with speeds of 4.77 m/s and 3.33 m/s respectively. The carts stick together after colliding head-on. Find the final speed.
Conservation of momentum:
(M1+M2)V = M1*V1-M2*V2
M1 = 4.66 kg
M2 = 3.00 kg
V1 = 4.77 m/s.
V2 = 3.33 m/s.
Solve for V.
To find the final speed of the carts after they stick together, we can use the principle of conservation of momentum.
The equation for conservation of momentum is given by:
(m1 * v1) + (m2 * v2) = (m1 + m2) * vf
where:
m1 and m2 are the masses of the carts,
v1 and v2 are the initial velocities of the carts,
and vf is the final velocity of the carts.
Given:
m1 = 4.66 kg,
m2 = 3.00 kg,
v1 = 4.77 m/s,
v2 = 3.33 m/s.
Substituting these values into the equation, we have:
(4.66 kg * 4.77 m/s) + (3.00 kg * 3.33 m/s) = (4.66 kg + 3.00 kg) * vf
Simplifying this equation, we get:
22.2742 kg*m/s + 9.99 kg*m/s = 7.66 kg * vf
32.2642 kg*m/s = 7.66 kg * vf
Now, we can solve for vf by dividing both sides of the equation by 7.66 kg:
vf = 32.2642 kg*m/s / 7.66 kg
vf ≈ 4.21 m/s
Therefore, the final speed of the two carts, after colliding and sticking together, is approximately 4.21 m/s.
To find the final speed after the collision, we can use the law of conservation of momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity:
Momentum = mass × velocity
Before the collision, the momentum of the first cart is calculated as follows:
Momentum1 = mass1 × velocity1 = 4.66 kg × 4.77 m/s = 22.1782 kg⋅m/s
Before the collision, the momentum of the second cart is calculated as follows:
Momentum2 = mass2 × velocity2 = 3.00 kg × (-3.33 m/s) = -9.99 kg⋅m/s (negative because the cart is moving in the opposite direction)
The total momentum before the collision is the sum of the individual momenta:
Total momentum before = Momentum1 + Momentum2 = 22.1782 kg⋅m/s + (-9.99 kg⋅m/s) = 12.1882 kg⋅m/s
After the collision, the carts stick together and move as one object with a combined mass of 4.66 kg + 3.00 kg = 7.66 kg. Let's assume the final velocity is Vf.
The momentum after the collision is equal to the momentum before the collision:
Total momentum after = Total momentum before
(mass1 + mass2) × Vf = 12.1882 kg⋅m/s
Solving for Vf:
Vf = total momentum before / (mass1 + mass2) = 12.1882 kg⋅m/s / 7.66 kg ≈ 1.59 m/s
Therefore, the final speed of the carts after sticking together is approximately 1.59 m/s.