In a shipping company distribution center, an open cart of mass 55.0 kg is rolling to the left at a speed of 5.30 m/s (see the figure (Figure 1) ). You can ignore friction between the cart and the floor. A 18.0 kg package slides down a chute that is inclined at 37 ∘ from the horizontal and leaves the end of the chute with a speed of 3.20 m/s . The package lands in the cart and they roll off together.

Question

In a shipping company distribution center, an open cart of mass 55.0 kg is rolling to the left at a speed of 5.30 m/s (see the figure (Figure 1) ). You can ignore friction between the cart and the floor. A 18.0 kg package slides down a chute that is inclined at 37 ∘ from the horizontal and leaves the end of the chute with a speed of 3.20 m/s . The package lands in the cart and they roll off together.

Answer 1

To find the final velocity of the cart and the package combined, we can use the principle of conservation of momentum. This principle states that the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting on the system.

The momentum of an object is given by the product of its mass and velocity. So, we can calculate the initial momentum of the cart and the package separately:

Initial momentum of the cart (before the package lands):
Pcart_initial = mass_of_cart * velocity_of_cart

Initial momentum of the package (before it lands):
Ppackage_initial = mass_of_package * velocity_of_package

Since there are no external forces acting on the system, the total momentum before the package lands is equal to the total momentum after the package lands, which can be represented by the equation:

Pcart_initial + Ppackage_initial = Pcart_final + Ppackage_final

We know the mass and initial velocity of the cart, the mass and initial velocity of the package, and we need to find the final velocity of the cart and the package combined (Vfinal).

Substituting the values into the equation:

(mass_of_cart * velocity_of_cart) + (mass_of_package * velocity_of_package) = (mass_of_cart + mass_of_package) * Vfinal

Now we can solve for Vfinal:

Vfinal = ((mass_of_cart * velocity_of_cart) + (mass_of_package * velocity_of_package)) / (mass_of_cart + mass_of_package)

Substituting the given values:

mass_of_cart = 55.0 kg
velocity_of_cart = 5.30 m/s
mass_of_package = 18.0 kg
velocity_of_package = 3.20 m/s

Vfinal = ((55.0 kg * 5.30 m/s) + (18.0 kg * 3.20 m/s)) / (55.0 kg + 18.0 kg)

Calculating the expression on the top and bottom of the fraction separately:
(289.5 + 57.6) kg*m/s / 73 kg

Adding the numerator and denominator:
347.1 kg*m/s / 73 kg

Simplifying the fraction:
Vfinal = 4.75 m/s (rounded to two decimal places)

Therefore, the final velocity of the cart and the package combined is 4.75 m/s.