The velocity function of particle at time \(t\), \(v=g(t)\) is given by:

\(v=\dfrac{ds}{dt}\)

Acceleration function, \(a=h(t)\) is given by:

\(a=\dfrac{dv}{dt}=\dfrac{d^2s}{dt^2}\)

An instantaneous velocity of a particle that moves along a straight line from a fixed point from a displacement function, \(s=f(t)\) can be determined by substituting the value of \(t\) in the velocity function:

\(v=\dfrac{ds}{dt}\)

Instantaneous acceleration, \(a\) of a particle moving along a straight line and passes through a fixed point can be determined from a velocity function, \(v=f(t)\) or a displacement function, \(s=f(t)\) by substituting the value of \(t\) into the acceleration function:

\(a=\dfrac{dv}{dt}=\dfrac{d}{dt}\left( \dfrac{ds}{dt} \right)=\dfrac{d^2s}{dt^2}\)

A particle moves along a straight line. Its displacement, \(s\) m, from the fixed point \(O\) is given by \(s=3+2t-t^2\), where \(t\) is time, in seconds, after it starts moving. Determine the velocity function, \(v\) and acceleration function, \(a\) of the particle.

Given the displacement function, \(s=3+2t-t^2\).

Then, the velocity function at time \(t\):

\(\begin{aligned} v&=\dfrac{ds}{dt} \\ v&=2-2t. \end{aligned}\)

The acceleration function at time \(t\):

\(\begin{aligned} a&=\dfrac{dv}{dt} \\ a&=-2. \end{aligned}\)

The velocity function of particle at time \(t\), \(v=g(t)\) is given by:

\(v=\dfrac{ds}{dt}\)

Acceleration function, \(a=h(t)\) is given by:

\(a=\dfrac{dv}{dt}=\dfrac{d^2s}{dt^2}\)

An instantaneous velocity of a particle that moves along a straight line from a fixed point from a displacement function, \(s=f(t)\) can be determined by substituting the value of \(t\) in the velocity function:

\(v=\dfrac{ds}{dt}\)

Instantaneous acceleration, \(a\) of a particle moving along a straight line and passes through a fixed point can be determined from a velocity function, \(v=f(t)\) or a displacement function, \(s=f(t)\) by substituting the value of \(t\) into the acceleration function:

\(a=\dfrac{dv}{dt}=\dfrac{d}{dt}\left( \dfrac{ds}{dt} \right)=\dfrac{d^2s}{dt^2}\)

A particle moves along a straight line. Its displacement, \(s\) m, from the fixed point \(O\) is given by \(s=3+2t-t^2\), where \(t\) is time, in seconds, after it starts moving. Determine the velocity function, \(v\) and acceleration function, \(a\) of the particle.

Given the displacement function, \(s=3+2t-t^2\).

Then, the velocity function at time \(t\):

\(\begin{aligned} v&=\dfrac{ds}{dt} \\ v&=2-2t. \end{aligned}\)

The acceleration function at time \(t\):

\(\begin{aligned} a&=\dfrac{dv}{dt} \\ a&=-2. \end{aligned}\)