Question

A motorboat starting from rest on a lake accelerates in a straight line at a constant rate of $3.0m/{{s}^{2}}$ for $8.0s$ . How far does the boat travel during this time?

Answer 1

Hint
Since the boat accelerates along the surface of the lake, we can call the motion as translational motion. For analysis of translational motion, we have Newton's laws of motion. We have been provided with the initial velocity of the motorboat, the acceleration of the motorboat, and the time of motion. So we can apply the second equation of motion to obtain our answer.
$\Rightarrow s=ut+\dfrac{1}{2}a{{t}^{2}}$

Complete step by step answer
Since the motorboat is starting from rest, we can say that
The initial velocity of the motorboat $(u)=0m/s$
We have been given that, the acceleration of the motorboat $(a)=3.0\ m/{{s}^{2}}$
The time for which the motion persists $(t)=8.0s$
From the second equation of motion given by Newton, we know that the displacement of a particle can be given as $(s)=ut+\dfrac{1}{2}a{{t}^{2}}$ where the meaning of the symbols has been discussed above
Substituting the values, we get
The displacement of the particle $(s)=0\times 8+\dfrac{1}{2}\times 3\times {{(8)}^{2}}$
Simplifying the above equation, we get
$\Rightarrow s=\dfrac{3\times 64}{2}=96m$
This is the distance the motorboat travels in the given time.
Note that the distance and the displacement of the motorboat would be the same as the boat is traveling in a straight line.

Note
Alternatively, to solve this question, we could make use of the first and the third equation of motion to get the same answer. We could find the final velocity using the initial velocity, the acceleration, and the time given; by applying the first equation of motion we know as $v=u+at$. Once the final velocity has been found, we can substitute the values of the initial velocity, the final velocity and the acceleration into the third equation of motion we know as ${{{v}}^{{2}}}{=}{{{u}}^{{2}}}{+2as}$ to obtain the required displacement.