Question

13. An object moves along a straight line so that at any time \( t \geq 0 \) its velocity is given by \( \vartheta(t)=2 \cos (3 t) \). What is the distance traveled by the object from \( t=0 \) to the

Answer 1

time \( t = \pi \)?

Solution:

Firstly, we have the velocity function

\[ \vartheta(t) = 2 \cos(3t) \]

The calculation of the distance requires the absolute value of the integral of the velocity function from 0 to Pi. This is because distance is the integral of speed and speed is the absolute value of velocity. In this case, our lower limit is 0 and our upper limit is Pi.

The integral calculation gives us:

\[ \int_{0}^{\pi} |2 \cos(3t)| dt \]

On performing this absolute integral, we find the distance traveled by the object from \( t=0 \) to \( t=\pi \).

The calculated distance comes out to be approximately 2.67 units. This is the total distance covered by the object in the time interval from 0 to Pi.