In the diagram above, a mass m starting at point A is...

In the diagram above, a mass $m$ starting at point $A$ is projected with the same initial horizontal velocity $v_{0}$ along each of the three tracks shown here (with negligible friction) sufficient in each case to allow the mass to reach the end of the track at point $B .$ (Path 1 is directed up, path 2 is directed horizontal, and path 3 is directed down.) The masses remain in contact with the tracks throughout their motions. The displacement $A$ to $B$ is the same in each case, and the total path length of path 1 and 3 are equal. If $t_{1}, t_{2},$ and $t_{3}$ are the total travel times between $A$ and $B$ for paths $1,2,$ and $3,$ respectively, what is the relation among these times? (A) $t_{3} < t_{2} < t_{1}$ (B) $t_{2} < t_{3} < t_{1}$ (C) $t_{2} < t_{1}=t_{3}$ (D) $t_{2}=t_{3} < t_{1}$

In the diagram above, a mass $m$ starting at point $A$ is projected with the same initial horizontal velocity $v_{0}$ along each of the three tracks shown here (with negligible friction) sufficient in each case to allow the mass to reach the end of the track at point $B .$ (Path 1 is directed up, path 2 is directed horizontal, and path 3 is directed down.) The masses remain in contact with the tracks throughout their motions. The displacement $A$ to $B$ is the same in each case, and the total path length of path 1 and 3 are equal. If $t_{1}, t_{2},$ and $t_{3}$ are the total travel times between $A$ and $B$ for paths $1,2,$ and $3,$ respectively, what is the relation among these times?
(A) $t_{3} < t_{2} < t_{1}$
(B) $t_{2} < t_{3} < t_{1}$
(C) $t_{2} < t_{1}=t_{3}$
(D) $t_{2}=t_{3} < t_{1}$

Key Concepts

Conservation of Mechanical Energy

This principle states that in the absence of friction or any non-conservative forces, the total energy (kinetic plus potential) of a system remains constant. As an object ascends or descends along a frictionless path, energy is converted between kinetic and gravitational potential energy, affecting its speed and influencing overall travel time.

Dynamics on Inclined Planes

When an object moves along an inclined surface, the gravitational force can be decomposed into components parallel and perpendicular to the plane. The parallel component affects the acceleration or deceleration of the object along the track, which plays a key role in determining the time required to travel a given distance.

Impact of Gravity on Motion

Gravity not only provides the force that accelerates an object downwards but also decelerates it when moving upward. This dual effect of gravity means that the direction of the inclined motion will affect the instantaneous speed of the object, thereby influencing the total travel time along different trajectories.

Path Length versus Displacement

Even if the overall displacement between two points is fixed, the actual path taken—characterized by its length and profile—can significantly affect the travel time. A path that effectively utilizes gravitational acceleration, even if it is longer, can result in a shorter travel time compared to a shorter, level, or rising path.

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