Final answer:
To calculate the minimum length L for a ramp to stop a truck, conservation of energy is used. Changing the truck's mass does not affect L, but decreasing the truck's speed reduces the required length L.
Step-by-step explanation:
To calculate the minimum length L that the ramp must have to stop the truck, we can use the principle of conservation of energy. The kinetic energy the truck has while moving at 172 km/h (which is approximately 47.78 m/s) needs to be completely converted into potential energy as it moves up the ramp.
The initial kinetic energy (KE) of the truck is given by the formula
KE = (1/2)mv²,
where m is the mass of the truck, and v is its velocity. The potential energy (PE) at the top of the ramp when the truck comes to a stop is given by PE = mgh, where g is the acceleration due to gravity, and h is the height the truck has climbed.
Without friction, all of the kinetic energy converts into potential energy: KE = PE, leading to
(1/2)mv² = mgh.
We can cancel m from both sides and solve for h, then use trigonometry to relate h to the length L of the ramp: sin(\theta) = h/L.
By substituting h and rearranging, we can solve for L.
As for the questions of how changing the mass or speed of the truck would affect the minimum length L: 'b' Decreasing the truck's mass would not affect L, because mass cancels out in the conversion from kinetic to potential energy. 'c' Decreasing the truck's speed would decrease the required length L because the initial kinetic energy would be lower.