2. A Particle moves in a straight line with its displacement of motion described by s(t) =ť² – 3t +5 where s is measured in feet and t is measured in seconds. Find the velocity of s at t = a.
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.
![### Problem Statement
A particle moves in a straight line with its displacement of motion described by the function:
\[ s(t) = t^2 - 3t + 5 \]
where \( s \) is measured in feet and \( t \) is measured in seconds.
**Question:**
Find the velocity of \( s \) at \( t = a \).
### Solution
To find the velocity of the particle at any time \( t \), you need to determine the derivative of the displacement function \( s(t) \) with respect to time \( t \). The derivative of \( s(t) \), denoted as \( s'(t) \), represents the velocity function \( v(t) \).
The displacement function given is:
\[ s(t) = t^2 - 3t + 5 \]
Taking the derivative with respect to \( t \):
\[
\begin{align*}
v(t) &= \frac{d}{dt}[t^2 - 3t + 5] \\
&= \frac{d}{dt}[t^2] - \frac{d}{dt}[3t] + \frac{d}{dt}[5] \\
&= 2t - 3 + 0
\end{align*}
\]
So the velocity function is:
\[ v(t) = 2t - 3 \]
To find the velocity at \( t = a \):
\[ v(a) = 2a - 3 \]
Hence, the velocity of the particle at \( t = a \) is \( 2a - 3 \) feet per second.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff60264e5-c449-444f-833d-dbd8ab630a5d%2F04f3d89f-285e-4192-9d16-42026070f9fd%2Fz3wptr_processed.png&w=3840&q=75)
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