Question

A hammer of mass $5kg$ moving with a speed of $2m{s^{ - 1}}$ strikes the head of a nail driving it $20cm$ into the wall. Find the impulse.
(A) $10Ns$
(B) $20Ns$
(C) $30Ns$
(D) $40Ns$

Answer 1

Large amount of force acted in a small amount of time is called impulse. Mathematically, it is given as the product of force and the small change in time.

Formula Used:
$I = F\Delta t$
Where,
$I$ is the impulse
$F$ is the force applied
$\Delta t$ is the time period for which the force is applied.

Complete step by step answer:
Momentum of any object is the product of mass and velocity of the object.
$\Rightarrow P = mv$
Thus, change in momentum can be given by
$\Delta P = m\Delta v$
Force, exerted on any object is given by the change in momentum of an object with respect to time.
$\Rightarrow F = \dfrac{{\Delta P}}{{\Delta t}}$
By cross multiplying, we get
$\Delta P = F\Delta t$ ………….. (1)
Now, by definition, impulse is the force acting on a body for a very short period of time.
i.e. impulse is the product of force and a small change in time.
$I = F\Delta t$ …………... (2)
Where,
$I$ is impulse
$F$ is force applied
$\Delta t$ small change in time
From equation (1) and (2), we get
$I = \Delta P = m\Delta v$
Now, when swung, it is its initial speed. i.e. $u = 2m{s^{ - 1}}$
And when the hammer strikes the nail, it will stop moving. So its final velocity will be zero. i.e. $v = 0$
It is given that the mass of the hammer is $m = 5kg$
Therefore, by using the above equation, we can write
$I = m(v - u)$
$= 5(0 - 2)$
$\Rightarrow I = 10Ns$

Hence, the correct option is (A), Impulse is 10Ns.

Note: Force is a vector quantity, therefore, impulse is also a vector quantity. This question is a good example that helps understand the difference, between force, momentum and impulse.