Final answer:
By using the conservation of momentum principle, the final velocity of the garbage truck after it collides with and adheres to a trash can is calculated to be approximately 24.9 m/s, slightly less than its initial velocity because of the additional mass of the trash can.
Step-by-step explanation:
To determine the speed of the garbage truck after it hits and adheres to a trash can, we can apply the principle of conservation of linear momentum. According to this principle, when no external forces act on a system, the total momentum of the system remains constant before and after a collision.
In this scenario, the garbage truck and the trash can system can be considered isolated since the scenario suggests ignoring any external factors like air resistance or friction. We can set up the equation for conservation of momentum as follows:
Momentum before collision = Momentum after collision
Mass of the truck × Initial velocity of the truck + Mass of the trash can × Initial velocity of the trash can = (Mass of the truck + Mass of the trash can) × Final velocity of the combined system
Substituting the given values:
(1.20 × 10⁴ kg) × (25.0 m/s) + (80.0 kg) × (0 m/s) = (1.20 × 10⁴ kg + 80.0 kg) × Final velocity
Final velocity = (1.20 × 10⁴ kg × 25.0 m/s)/(1.2008 × 10⁴ kg)
Calculating this, the final velocity of the garbage truck after the collision is:
Final velocity ≈ 24.9 m/s
The small decrease in velocity compared to the initial speed of the truck occurs due to the additional mass of the trash can combined with the truck after the collision.