A particle's acceleration is described by the function a(t) = (12 t-30) m/s?, where t is in s. Its initial conditions are xo = 0 m and vo = 0 m/s at t = 0 s. a. At what time is the velocity is 3m/s? b. What is the particle's position at that time?
Displacement, Velocity and Acceleration
In classical mechanics, kinematics deals with the motion of a particle. It deals only with the position, velocity, acceleration, and displacement of a particle. It has no concern about the source of motion.
Linear Displacement
The term "displacement" refers to when something shifts away from its original "location," and "linear" refers to a straight line. As a result, “Linear Displacement” can be described as the movement of an object in a straight line along a single axis, for example, from side to side or up and down. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Linear displacement is usually measured in millimeters or inches and may be positive or negative.
The particle's acceleration is described by
--------(1)
Given initial conditions are
Since we know that the acceleration is given by
Integrating both sides
where C is the constant of integration
Here we use
--------(2)
From initial condition, we know at
Using initial condition in (2), we get
Therefore, equation 2 becomes
--------(3)
Let us find the find the time at which v=3m/s. Substitute v=3m/s in (3) and solve t
This is a quadratic equation in t
for solving quadratic equation we use
Since the time cannot be negative, we neglect the negative value of t
Therefor, the particle's velocity is 3m/s at 5.1s.
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