Question

the velocity of a particle moving along the x-axis is given by ( f(t)=9 - 3t ) cm/sec. use a graph of ( f(t) ) to find the exact change in position of the particle from time ( t = 0 ) to ( t = 4 ) seconds. change in position = (include units)

Explanation:

Step1: Find the \(t -\)intercept of \(y = f(t)\)

Set \(f(t)=0\), so \(9 - 3t=0\). Solving for \(t\) gives \(t = 3\).

Step2: Calculate the area of the regions

The velocity - time graph \(y=f(t)=9 - 3t\) is a straight line.
For \(0\leq t\leq3\), the region between the curve \(y = f(t)\) and the \(t -\)axis is a triangle. The base \(b = 3\) and the height \(h=f(0)=9\). The area \(A_1=\frac{1}{2}\times3\times9=\frac{27}{2}\).
For \(3\lt t\leq4\), the region between the curve \(y = f(t)\) and the \(t -\)axis is a triangle. The base \(b'=4 - 3 = 1\) and the height \(h'=\vert f(4)\vert=\vert9-3\times4\vert = 3\). The area \(A_2=\frac{1}{2}\times1\times3=\frac{3}{2}\).

Step3: Compute the net change in position

The net change in position (displacement) is \(A=A_1 - A_2\).
Substitute \(A_1\) and \(A_2\) into the formula: \(A=\frac{27}{2}-\frac{3}{2}=\frac{27 - 3}{2}=12\).

Answer:

\(12\space cm\)