Question

A hammer of mass $1kg$ strikes on the head of a nail with a velocity of $2m{{s}^{-1}}$ . It drives the nail $0.01m$ into a wooden block. Find the force applied by the hammer and the time of impact.
A) $200N;{{10}^{-2}}s$
B) $100N;{{10}^{-1}}s$
C) $300N;{{10}^{-2}}s$
D) $400N;{{10}^{-3}}s$

Answer 1

Equations of motion can be used to find the force and the time of impact. The third equation of motion can be used to find the acceleration and thus, the force applied by the hammer. The first equation of motion can be used to find the time of impact.

Formula used:
$1){{v}^{2}}-{{u}^{2}}=2as$
where
$v$ is the final velocity
$u$ is the initial velocity
$a$ is the acceleration
$s$ is the distance covered

$2)v=u+at$
where
$v$ is the final velocity
$u$ is the initial velocity
$a$ is the acceleration
$t$ is the time required

Complete answer:
We are given that a hammer is used to strike on the head of a nail to drive it to a wooden block. The mass of the hammer is given as $m=1kg$ and the velocity with which it hits the nail is given as $2m{{s}^{-1}}$. We are also provided that the nail is driven by a distance of $0.01m$ into a wooden block. We are required to find the force with which the hammer hits the nail and the time of impact for the nail to drive into the wooden block.
From the third equation of motion, we have
${{v}^{2}}-{{u}^{2}}=2as$
where
$v$ is the final velocity
$u$ is the initial velocity
$a$ is the acceleration
$s$is the distance covered
Let this be equation 1.
Applying equation 1 to the question provided, we have
${{v}^{2}}-{{u}^{2}}=2as\Rightarrow {{0}^{2}}-{{2}^{2}}=2\times a\times 0.01$
where
$v$ is the final velocity of the hammer$(=0m{{s}^{-1}})$
$u$ is the initial velocity of the hammer$(=2m{{s}^{-1}})$
$a$ is the acceleration of the hammer
$s$is the distance covered by the nail as a consequence of the force applied by the hammer$(=0.01m)$
Let this be equation 2. Note that the final velocity of the hammer is equal to zero because the hammer comes to a stop after hitting the nail.
Simplifying equation 2, we have
${{0}^{2}}-{{2}^{2}}=2\times a\times 0.01\Rightarrow -4=0.02a\Rightarrow a=-200m{{s}^{-2}}$
Therefore, the magnitude of acceleration of the hammer is equal to $200m{{s}^{-2}}$.
Let this be equation 3.
Now, the magnitude of force of the hammer is given by
$F=ma=1kg\times 200m{{s}^{-2}}=200N$
Let this be equation 4.
Moving on, let us calculate the time of impact for the hammer to drive the nail into the wooden block. This can be obtained using the first equation of motion, which states that
$v=u+at$
where
$v$ is the final velocity
$u$ is the initial velocity
$a$ is the acceleration
$t$ is the time required
Applying this equation here, we have
$v=u+at\Rightarrow 0=2+200(t)$
where
$v$ is the final velocity of the hammer$(=0m{{s}^{-1}})$
$u$ is the initial velocity of the hammer$(=2m{{s}^{-1}})$
$a$ is the acceleration of the hammer$(=200m{{s}^{-2}})$
Let this be equation 5.
On further simplification of equation 5, we have
$0=2+200(t)\Rightarrow t=-{{10}^{-2}}s$
Therefore, the time of impact is equal to ${{10}^{-2}}s$.
Let this be equation 6.

From equation 4 and equation 6, it is clear that the hammer hits the nail with a force of $200N$ and the time required for the nail to drive into the wooden block is ${{10}^{-2}}s$.

So, the correct answer is “Option A”.

Note:
Students need not get confused with the signs of acceleration, force and time of impact. We are supposed to find the magnitudes only. The reason behind these negative signs is that the hammer is coming to a stop on the application of force to the nail. The change in velocity in this process is negative as the final velocity is zero. The acceleration in this case is called retardation. Since change in velocity is negative, acceleration turns out to be negative, which further affects the sign of applied force as well as the time of impact.