Question

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

Answer 1

When a body is said to move with a constant acceleration in a straight line then the laws and equations of kinematics come into picture. The problem requires the identification of which of the three equations of kinematics are to be applied. The straight line motion equation is applied in order to determine the required quantity.

Formula used:
The equation for the straight line of motion is given by:
$s = u{t^2} + \dfrac{1}{2}a{t^2}$
Where, $s$ is the distance covered, $u$ is the initial velocity, $t$ is the time taken and $a$ is the acceleration.

Complete step by step answer:
The problem revolves around the concept of a body moving in a straight line motion which is described by the equations of kinematics. Kinematics is a branch of physics wherein the concept speaks about the motion of a body with respect to different dimensions like one dimension, two dimension etc. and the external forces which may be applied on them are not considered.

From the above question, it is clear that the boat is said to exhibit one dimensional motion because the motion of the boat is observed to be only in one direction. The motion of the body is said to be one dimensional when only one of the three quantities which specify the position of the object varies with time.

Here, the boat is said to travel across the lake and to get to the other side it needs to travel in a straight line motion which means that only one of the coordinates indicating the position of the boat will vary with time. Acceleration is said to be the rate of change of velocity with time. In simple words when motion is constant with a variation in velocity and its corresponding increase in speed then a body is said to accelerate and gain velocity. Here, we consider uniform acceleration with time and hence velocity changes linearly with time.

When a body is said to exhibit a uniformly accelerated motion along a straight line then three equations for kinematics were derived. Out of these three equations we are going to apply the following equation as per the requirement of this problem:
$s = u{t^2} + \dfrac{1}{2}a{t^2}$ -------($1$)
Let us now extract the data given in the question. We are asked to find out how far the boat will travel which means we need to determine the distance covered by the boat. The acceleration of the boat and the time taken for it to travel a certain distance is given. The boat is said to accelerate only after a certain interval of time.

Given, $a = 3.0\;m/{s^2}$ and $t = 8\;s$. By the concept of rest and motion, a body is said to be at rest when it does not change its position with respect to time. In the question it is given that at the start, that is, initially before the boat started to accelerate the boat was at rest which means that the initial velocity of the boat will be zero since there was no motion and hence no velocity. Hence, we can say that:
$u = 0$
By substituting the above condition we get:
$s = \left( 0 \right){t^2} + \dfrac{1}{2}a{t^2}$
$\Rightarrow s = 0 + \dfrac{1}{2}a{t^2}$
By substituting the given values into the equation ($1$) we get:
$s = \dfrac{1}{2}\left( 3 \right){\left( 8 \right)^2}$
$\Rightarrow s = \dfrac{{192}}{2}$
$\therefore s = 96\;m$

Hence the boat must travel a distance of $96\;m$ to get to the other side of the lake.

Note: The issue faced in these types of problems would be to determine which of the three equations of motion are to be applied when. A common mistake may be the usage of the wrong equation and hence the equations are applied in accordance to the quantity which is asked. The first equation of motion gives the speed of the body over a time period, the second equation gives the position of the body and the third equation specifies the speed of the body over some distance.