Question

A constant retarding force of \[50\;{\text{N}}\] is applied to a body of mass \[{\text{10}}\;{\text{kg}}\] moving initially with a speed of \[{\text{10}}\;{\text{m/s}}\]. How long the body takes to comes to rest?
A. \[{\text{2}}\;{\text{s}}\]
B. \[{\text{4}}\;{\text{s}}\]
C. \[{\text{6}}\;{\text{s}}\]
D. \[{\text{8}}\;{\text{s}}\]

Answer 1

Hint: Formula for force is given by,
\[F = ma\]. Obtain the value of acceleration from the equation of force.
According to laws of motion,
\[v = u + at\]. Obtain the value of required time, \[t\] from that equation.

Complete step by step solution:
Given,
The force is, \[F = - 50\;{\text{N}}\], (the applied force is a retarding force, hence we use negative sign before the numerical value of the force)
Mass of the body is given by, \[m = {\text{10}}\;{\text{kg}}\]
Initial speed of the body is given by, \[u = {\text{10}}\;{\text{m/s}}\]
Here, final speed of the body is, \[v = 0\;{\text{m/s}}\]
Now, we need to find the value of time, \[t\].

We know that, the formula for force is deduced by Newton’s law, and is given by
\[F = ma\], …… (i)
Where \[F\] is force,
\[m\] is mass of a body, and
\[a\] is acceleration of the body.

In order to find the acceleration we rearrange the equation (i) as,
\[a = \dfrac{F}{m}\] …… (ii)
Now substitute the value of \[F = - 50\;{\text{N}}\] and \[m = {\text{10}}\;{\text{kg}}\] in equation (ii).
Therefore,
\[\begin{gathered}
  a = \dfrac{{ - 50\;{\text{N}}}}{{10\;{\text{kg}}}} \\
   = - 5\;{\text{m/}}{{\text{s}}^{\text{2}}} \\
\end{gathered} \]

Now, in order to find the required time we use the formula of motion, and is given by,
\[v = u + at\]
Rearrange the above equation to find the value of \[t\].
\[t = \dfrac{{v - u}}{a}\] …… (iii)
Now, substitute the values of \[u = {\text{10}}\;{\text{m/s}}\], \[v = 0\;{\text{m/s}}\], and \[a = - 5\;{\text{m/}}{{\text{s}}^{\text{2}}}\] in equation (iii)
Therefore,
\[\begin{gathered}
  t = \dfrac{{0\;{\text{m/s}} - 1{\text{0}}\;{\text{m/s}}}}{{ - 5\;{\text{m/}}{{\text{s}}^{\text{2}}}}} \\
   = \dfrac{{10}}{5}\;{\text{s}} \\
   = {\text{2}}\;{\text{s}} \\
\end{gathered} \]

Hence, the required time is \[2\;{\text{s}}\].

Note: In this problem we are asked to calculate the value of time taken. For this, use the formula
\[v = u + at\] instead of \[S = ut + \dfrac{1}{2}a{t^2}\] or \[{v^2} - {u^2} = 2aS\]. But we need the value of \[a\] to solve it. Now in order to find the value of \[a\], use the formula \[F = ma\]. Subtract \[10\;{\text{m/s}}\] from \[0\;{\text{m/s}}\] to get the correct answer or else the time will come as negative.