In kinematics it may be necessary to analyze a situation in which particles do not exhibit regular displacement throughout their movement, so that we cannot use just a mathematical equation to describe their velocity, acceleration and displacement. In cases like these, we use graphs to represent the motion of a particle, since the fundamental equations of speed and acceleration have geometric relationships with the graphs that represent them.
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.
In
Regarding the graphs of acceleration as a function of time, position as a function of time and speed as a function of time, it can be stated that:
Choose one:
The. The difference between velocities v2 and v1 is numerically equal to the area under the position curve between time t1 and t2, and therefore can be calculated by the derivative: v2-v1 = dS / dt. The difference between the accelerations a1 and a2 is numerically equal to the area under the velocity curve between the time t1 and t2 and can be calculated by the derivative: a2-a1 = dv / dt.
B. The difference between positions x2 and x1 is numerically equal to the area under the velocity curve between time t1 and t2, and therefore can be calculated by the integral: x2-x1 = ∫vdt. The difference between speeds v1 and v2 is numerically equal to the area under the acceleration curve between time t1 and t2 and can be calculated by the integral: v2-v1 = ∫adt.
ç. The instantaneous speed is calculated using the derivative of the position as a function of time and is represented by the slope of the velocity curve as a function of time. The instantaneous acceleration is calculated by the derivative of the speed as a function of time, being represented by the slope of the acceleration curve as a function of time.
d. The difference between positions x2 and x1 is numerically equal to the area under the acceleration curve between time t1 and t2, and therefore can be calculated by the integral: x2-x1 = ∫adt. The difference between speeds v1 and v2 is numerically equal to the area under the position curve between time t1 and t2 and can be calculated by the integral: v2-v1 = ∫xdt.
and. The instantaneous speed is calculated through the derivative of acceleration as a function of time and is represented by the slope of the velocity curve as a function of time. The instantaneous acceleration is calculated by the second derivative of the position as a function of time, being represented by the slope of the acceleration curve as a function of time.
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