Two carts with masses of 4.20 kg and 2.44 kg move toward each other on a frictionless track with speeds of 4.92 m/s and 3.87 m/s respectively. The carts stick together after colliding head-on. Find the final speed.
To find the final speed of the carts after the collision, we can apply the principle of conservation of momentum.
According to the principle of conservation of momentum, 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. Mathematically, momentum (p) is defined as:
p = m * v
Where:
p = momentum
m = mass of the object
v = velocity of the object
Let's denote the initial velocity of the first cart as v1, the initial velocity of the second cart as v2, and the final velocity of the combined carts as v.
Now we can calculate the momentum before the collision:
Initial momentum of the first cart = m1 * v1
Initial momentum of the second cart = m2 * v2
Since the carts are moving in opposite directions, the momentum of the first cart is negative.
So, the net initial momentum before the collision is:
Initial momentum = -m1 * v1 + m2 * v2
According to the conservation of momentum principle, the final momentum is equal to the initial momentum:
Final momentum = (m1 + m2) * v
Now, equating the initial and final momentum, we can solve for the final velocity (v).
-m1 * v1 + m2 * v2 = (m1 + m2) * v
Substituting the given values:
-m1 * 4.92 + m2 * 3.87 = (m1 + m2) * v
Using the masses given:
-(4.20 kg) * 4.92 m/s + (2.44 kg) * 3.87 m/s = (4.20 kg + 2.44 kg) * v
Simplifying the equation:
-20.71 kg*m/s + 9.44 kg*m/s = 6.64 kg * v
-11.27 kg*m/s = 6.64 kg * v
Dividing both sides by 6.64 kg:
v = -11.27 kg*m/s / 6.64 kg
v ≈ -1.70 m/s
Since the negative sign indicates that the carts are moving in opposite directions after the collision, we can drop the negative sign and take the magnitude of the velocity.
Thus, the final speed of the combined carts after the collision is approximately 1.70 m/s.