Question

A motor boat starting from rest on a lake accelerates in a straight line at a constant rate of $3{\text{ m/}}{{\text{s}}^2}$ for 8 seconds. How far does the boat travel during this time?

Answer 1

In the given question we have to find the distance that the boat has covered. The initial velocity, acceleration and time taken by the boat are given. To find the distance covered we are going to second equation of motion which is given below:
$s = ut + \dfrac{1}{2}a{t^2}$

Complete step by step solution:
Given,
$a = 3{\text{ m/}}{{\text{s}}^2}$
$t = 8s$
As the boat is starting from rest so initial velocity will be equal to zero, $u = 0$
Let the distance covered by the boat is s, then by second equation of motion the distance covered is given by following formula,
$\Rightarrow s = ut + \dfrac{1}{2}a{t^2}$
Putting the values of acceleration, initial velocity and time taken.
$\Rightarrow s = 0 \times 8 + \dfrac{1}{2} \times 3 \times {8^2}$
$\Rightarrow s = 0 + \dfrac{{192}}{2}$
$\Rightarrow x = \dfrac{{192}}{2}$
$\Rightarrow s = 96{\text{ m}}$

Hence, from above calculation we have got the value of the distance covered by the boat $s = 96{\text{ m}}$.

Note: In this question to find the value of the distance covered by the boat we have used Newton’s equations of motion. There are a total three equations of motion. These equations are given below:
$v = u + at$
$s = ut + \dfrac{1}{2}a{t^2}$
${v^2} = {u^2} + 2as$
Where,
$v$ is final velocity in $m/s$
$u$ is initial velocity in $m/s$
$a$ is acceleration in ${\text{m/}}{{\text{s}}^2}$
$t$ is time in $s$
$s$ is the distance in $m$
These equations have a great significance in Physics as they are used to find different variables in different conditions. These equations are independent of the mass of the object. These equations describe the behavior of that system with respect to time. These equations of motion give information of a system in terms of a mathematical function. So, by looking at the equations one can understand the system. If there is force acting on a body then it will be difficult to find the equation of motion of that system. In such conditions we use other methods to find the equation of motion of that particular system.