A 1400 kg car is moving westward with a speed of 12 m/s, and a 3600 kg truck is traveling east with a speed of 27 m/s. Find the velocity of the center of mass of the system. Answer in units of m/sˆ
Center of mass is:
(m1x1+m2x2+...)/(m1+m2+...)
Therefore, the center of mass is:
(1400*-12 + 3600*27)/(1400+3600)
or 16.08 m/s
Well, since the car is moving westward and the truck is moving eastward, it seems like they want to have a little rendezvous in the middle. It's like a game of "meet me halfway" between the car and the truck.
To find the velocity of the center of mass, we need to consider the masses and velocities of both the car and the truck. Let's call the velocity of the car "Vc" and the velocity of the truck "Vt".
Since the car has a mass of 1400 kg and a velocity of 12 m/s to the west, we can say that the momentum of the car is given by: Momentum_c = mass_c * Vc
= 1400 kg * (-12 m/s) [negative because it's moving west]
= -16800 kg.m/s
Similarly, the momentum of the truck can be found using its mass of 3600 kg and velocity of 27 m/s to the east: Momentum_t = mass_t * Vt
= 3600 kg * 27 m/s
= 97200 kg.m/s
Now, since momentum is a vector quantity, which means it has both magnitude and direction, we need to combine the momenta of the car and truck to find the overall momentum of the system.
The total momentum of the system (car + truck) is given by: Momentum_total = Momentum_c + Momentum_t
= -16800 kg.m/s + 97200 kg.m/s
= 80400 kg.m/s
Finally, to find the velocity of the center of mass, we divide the total momentum by the total mass of the system (car + truck): Velocity_cm = Momentum_total / (mass_c + mass_t)
= 80400 kg.m/s / (1400 kg + 3600 kg)
= 80400 kg.m/s / 5000 kg
= 16.08 m/s
So, the velocity of the center of mass of the system is approximately 16.08 m/s. It's like they found the perfect speed to go on a date in the middle!
To find the velocity of the center of mass of the system, we need to use the principle of conservation of momentum. The formula for the velocity of the center of mass is:
Vcm = (m1 * v1 + m2 * v2) / (m1 + m2)
Where:
- Vcm is the velocity of the center of mass
- m1 and m2 are the masses of the objects
- v1 and v2 are the velocities of the objects
Given:
- Car mass (m1) = 1400 kg
- Car velocity (v1) = 12 m/s (westward)
- Truck mass (m2) = 3600 kg
- Truck velocity (v2) = 27 m/s (eastward)
Substituting the given values into the formula, we have:
Vcm = (1400 kg * 12 m/s + 3600 kg * (-27 m/s)) / (1400 kg + 3600 kg)
Simplifying the equation:
Vcm = (16800 kg·m/s - 97200 kg·m/s) / 5000 kg
Vcm = (- 80400 kg·m/s) / 5000 kg
Vcm = - 16.08 m/s
Therefore, the velocity of the center of mass of the system is -16.08 m/s, directed westward.
To find the velocity of the center of mass of the system, we need to consider the conservation of momentum.
The momentum of an object is given by the product of its mass and velocity. Mathematically, it can be expressed as:
Momentum = mass × velocity
The total momentum of the system before the collision is equal to the total momentum of the system after the collision. Since there is no external force acting on the system, the total momentum is conserved.
Let's assume the positive direction is towards the east. The momentum of the car before the collision is calculated as the product of its mass and velocity, which is:
Momentum_car = mass_car × velocity_car
= 1400 kg × (-12 m/s)
= -16,800 kg⋅m/s
Here, we have assigned a negative sign to the velocity of the car since it is moving in the opposite direction of the assumed positive direction.
Similarly, the momentum of the truck before the collision is calculated as:
Momentum_truck = mass_truck × velocity_truck
= 3600 kg × 27 m/s
= 97,200 kg⋅m/s
The total momentum before the collision is the sum of the individual momenta:
Total momentum before = Momentum_car + Momentum_truck
= -16,800 kg⋅m/s + 97,200 kg⋅m/s
= 80,400 kg⋅m/s
Since momentum is conserved, the total momentum after the collision is also 80,400 kg⋅m/s.
The center of mass velocity of the system can be obtained by dividing the total momentum by the total mass of the system. The total mass is the sum of the masses of the car and the truck:
Total mass = mass_car + mass_truck
= 1400 kg + 3600 kg
= 5000 kg
Velocity of the center of mass = Total momentum / Total mass
= 80,400 kg⋅m/s / 5000 kg
= 16.08 m/s
Therefore, the velocity of the center of mass of the system is 16.08 m/s to the east.
Note: The sign of the velocity indicates the direction, with positive being towards the east and negative being towards the west.