Three carts of masses 5 kg, 12 kg, and 3 kg
move on a frictionless horizontal track with
speeds of 5 m/s, 8 m/s, and 4 m/s, as shown in
figure. The carts stick together after colliding.
5 kg 12 kg 3 kg
5 m/s 8 m/s 4 m/s
Find the final velocity of the three carts.
Answer in units of m/s
To find the final velocity of the three carts, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) is given by the equation:
p = m * v
Where p is the momentum, m is the mass, and v is the velocity.
Before the collision, each cart has its own momentum. Let's calculate the momentum of each cart separately:
For the 5 kg cart:
Momentum (p1) = mass (m1) * velocity (v1)
p1 = 5 kg * 5 m/s
For the 12 kg cart:
Momentum (p2) = mass (m2) * velocity (v2)
p2 = 12 kg * 8 m/s
For the 3 kg cart:
Momentum (p3) = mass (m3) * velocity (v3)
p3 = 3 kg * 4 m/s
Next, we need to calculate the total initial momentum before the collision. We simply add up the individual momenta of each cart:
Total initial momentum (P_initial) = p1 + p2 + p3
After the collision, the three carts stick together and move with a common final velocity, let's call it v_final.
To calculate the final velocity, we can use the principle of conservation of momentum:
Total initial momentum (P_initial) = Total final momentum (P_final)
Since the carts stick together and move with the same final velocity, we can express the final momentum as the sum of the masses multiplied by the final velocity:
P_final = (m1 + m2 + m3) * v_final
Setting the initial and final momenta equal, we have:
P_initial = P_final
p1 + p2 + p3 = (m1 + m2 + m3) * v_final
Now we can substitute the values we have:
(5 kg * 5 m/s) + (12 kg * 8 m/s) + (3 kg * 4 m/s) = (5 kg + 12 kg + 3 kg) * v_final
Simplifying the equation:
25 kg*m/s + 96 kg*m/s + 12 kg*m/s = 20 kg * v_final
133 kg*m/s = 20 kg * v_final
Divide both sides of the equation by 20 kg:
6.65 m/s = v_final
Therefore, the final velocity of the three carts after the collision is 6.65 m/s.