A truck loaded with sand accelerates along a highway. The driving force on the truck remains constant. What happens to the acceleration of the truck as its trailer leaks sand at a constant rate through a hole in its bottom?
F = m a
assume same F from tires on road
so
as m decreases, a increases
When the trailer of the truck leaks sand at a constant rate, the mass of the truck decreases over time. As a result, the net force acting on the truck decreases because the force required to move the smaller mass is reduced. Since the driving force on the truck remains constant, a smaller net force means that the acceleration of the truck decreases.
To mathematically understand this, let's consider Newton's second law of motion: F = ma, where F is the net force, m is the mass, and a is the acceleration.
Initially, when the truck is fully loaded with sand, let's say its mass is M. The net force (F_initial) acting on the truck is equal to the driving force.
F_initial = ma_initial
While the truck is moving, sand leaks out of the trailer at a constant rate, reducing the mass of the truck over time. Let's say at a certain moment, the mass of the truck has reduced to m, where m < M. The net force (F_final) acting on the truck now becomes:
F_final = ma_final
Since the driving force remains constant, F_initial = F_final. Therefore, we have:
ma_initial = ma_final
As the mass of the truck decreases (m < M), the acceleration of the truck (a_final) will also decrease compared to its initial acceleration (a_initial). Hence, the truck's acceleration decreases as the sand leaks from the trailer at a constant rate.
As the truck's trailer leaks sand at a constant rate through a hole in its bottom, the total mass of the truck and the sand inside the trailer decreases over time. According to Newton's second law of motion, the acceleration of an object is directly proportional to the net force and inversely proportional to the mass of the object.
Since the driving force on the truck remains constant, but the mass of the truck decreases, the net force acting on the truck (which is the driving force minus the force due to the leaking sand) decreases as well. As a result, the acceleration of the truck decreases over time.
To summarize, as the sand leaks out of the truck's trailer, the decreasing mass of the truck leads to a decrease in the net force acting on it, ultimately resulting in a decrease in its acceleration.