A railroad car (m=3000 kg) is coasting along a level track with an initial speed of 22 m/s. A load of coal is then dropped into the car as sketched in the figure below.
(a) What are the velocities of the car and the coal before the collision along the x direction?
Vcar=____m/s
Vcoal=____m/s
(b) If the final speed of the car plus coal is 17 m/s, what is the mass of the coal?
_____kg
To solve this problem, we will use the law of conservation of momentum. According to this law, the total momentum before the collision should be equal to the total momentum after the collision.
(a) Let's calculate the velocities of the car and the coal before the collision along the x-direction.
Initially, the car is coasting with a velocity of 22 m/s. Therefore, the initial velocity of the car is Vcar = 22 m/s.
Before the collision, the coal is at rest (assuming it drops vertically into the car). Thus, the initial velocity of the coal is Vcoal = 0 m/s.
(b) To find the mass of the coal, we need to consider the conservation of momentum after the collision. The final speed of the car plus the coal is given as 17 m/s. Therefore, the final velocity of the car plus the coal is Vfinal = 17 m/s.
According to the law of conservation of momentum, the total momentum before the collision must be equal to the total momentum after the collision:
Initial momentum = Final momentum
The initial momentum is the product of the car's mass and its initial velocity:
Initial momentum = mcar * Vcar
Similarly, the final momentum is the sum of the car's mass (mcar) and the mass of the coal (mcoal) multiplied by their final velocity (Vfinal):
Final momentum = (mcar + mcoal) * Vfinal
Setting the initial momentum equal to the final momentum:
mcar * Vcar = (mcar + mcoal) * Vfinal
Now, we have two unknowns: Vcar and mcoal. To solve for mcoal, we need one more equation.
We can use the conservation of mass, which states that the total mass before the collision is equal to the total mass after the collision:
Initial mass = Final mass
The initial mass includes only the car's mass, mcar, while the final mass includes the car's mass, mcar, and the mass of the coal, mcoal:
Initial mass = mcar
Final mass = mcar + mcoal
Setting the initial mass equal to the final mass:
mcar = mcar + mcoal
Subtracting mcar from both sides:
0 = mcoal
Since mcoal = 0, this implies that the mass of the coal is zero. This result seems unusual, indicating that either a mistake has been made in the problem or some relevant information is missing.
Therefore, the mass of the coal cannot be determined using the given information.