An object in rectilinear motion is moving along a horizontal line with velocity $v ( t ) = \sin ( t )$ m/s for the time interval $0 \leq t \leq \frac { 3 \pi } { 4 }$ seconds. What is the net displacement from $t = 0$ to $t = \frac { 3 \pi } { 4 }$ ?

A) $\frac { 7 \sqrt { 2 } + 16 } { 12 }$
B) $\frac { \sqrt { 2 } } { 4 }$ m
C) $\frac { \sqrt { 2 } - 8 } { 12 }$ m
D) $\frac { \sqrt { 2 } + 2 } { 2 }$ m
E) $\frac { \sqrt { 2 } + 8 } { 12 }$ m

Question

An object in rectilinear motion is moving along a horizontal line with velocity

v ( t ) = \sin ( t )

m/s for the time interval

0 \leq t \leq \frac { 3 \pi } { 4 }

seconds. What is the net displacement from

t = 0

to

t = \frac { 3 \pi } { 4 }

?

A) $\frac { 7 \sqrt { 2 } + 16 } { 12 }$
B) $\frac { \sqrt { 2 } } { 4 }$ m
C) $\frac { \sqrt { 2 } - 8 } { 12 }$ m
D) $\frac { \sqrt { 2 } + 2 } { 2 }$ m
E) $\frac { \sqrt { 2 } + 8 } { 12 }$ m

Quizplus · Accepted Answer

The net displacement of an object in rectilinear motion is given by the integral of its velocity function over the given time interval. Here, the velocity function is $v(t) = \sin(t)$, and the time interval is from $0$ to $\frac{3\pi}{4}$ seconds. The displacement, $S$, can be calculated as:$$S = \int_{0}^{\frac{3\pi}{4}} \sin(t) \, dt = [-\cos(t)]_{0}^{\frac{3\pi}{4}} = -\cos\left(\frac{3\pi}{4}\right) - (-\cos(0)) = -\left(-\frac{\sqrt{2}}{2}\right) - (-1) = \frac{\sqrt{2}}{2} + 1 = \frac{\sqrt{2} + 2}{2}$$ m.

An Object in Rectilinear Motion Is Moving Along a Horizontal v(t)=sin⁡(t)v ( t ) = \sin ( t )v(t)=sin(t)

An Object in Rectilinear Motion Is Moving Along a Horizontal $v ( t ) = \sin ( t )$