Statement:
The acceleration of a particle is given by ( a(t) = 12t - 6 ). At ( t=0 ), ( v_0 = 5 , \textm/s ), ( s_0 = 2 , \textm ). Find:
Solution:
1. Velocity:
( v(t) = \int a , dt = \int (12t - 6) dt = 6t^2 - 6t + C )
Using ( v(0) = 5 ): ( C = 5 )
( v(t) = 6t^2 - 6t + 5 )
2. Position:
( s(t) = \int v , dt = \int (6t^2 - 6t + 5) dt = 2t^3 - 3t^2 + 5t + D )
Using ( s(0) = 2 ): ( D = 2 )
( s(t) = 2t^3 - 3t^2 + 5t + 2 ) rectilinear motion problems and solutions mathalino upd
3. Max velocity:
Set ( a(t) = 0 ) → ( 12t - 6 = 0 ) → ( t = 0.5 , \texts )
Check second derivative of ( v ): ( v'(t) = a(t) ), ( a'(t) = 12 > 0 ) → minimum actually (since concave up)
Wait — ( a(t) = 12t - 6 ), derivative of ( a ) = 12 > 0 → acceleration increasing, so ( v ) has minimum at ( t=0.5 ).
Thus, no maximum for ( t \ge 0 ) — velocity increases indefinitely. So answer: no max (or infinite).
Answers:
( v(t) = 6t^2 - 6t + 5 )
( s(t) = 2t^3 - 3t^2 + 5t + 2 )
No finite maximum velocity.
Rectilinear Motion, or translational motion in a straight line, is one of the fundamental topics in Engineering Mechanics (Dynamics). In the context of the Mathalino curriculum and board exams, the focus is not just on memorizing formulas, but on identifying which "type" of motion problem is being presented. Statement: The acceleration of a particle is given
Here is a breakdown of the problem types, formulas, and sample solutions.
As he refreshed the page to check another problem, something was different. At the top of the page, a banner appeared:
[UPD] – Unified Problem Derivatives: New Rectilinear Motion Module Live
Includes time-varying acceleration, piecewise motion, and interactive velocity-time graph simulations. Solution: 1
“UPD” – he thought first of “University of the Philippines Diliman,” his own campus. But here, it meant Update. And this was not a minor fix. Mathalino had just released a major overhaul of its rectilinear motion section.
Curious, Miguel clicked the link. The new page featured:
One new problem caught his eye: Problem 1127 (UPD) – “A stone is thrown vertically upward from a cliff 100 m high with a speed of 30 m/s. On its way down, it just misses the thrower. Find the total time of flight and the velocity just before hitting the ground.”
The old Mathalino would solve it in two lines. The new version showed three methods: energy conservation, piecewise displacement, and symmetry of motion—then compared the results. A note read: “Many students forget that rectilinear motion includes vertical motion under gravity. Always define your positive direction and stick to it.”