Question

A bullet with mass m is projected out horizontally into the sand with speed v0. Assume it is subject to the resistance = (k is a constant). Ignore the gravity of the bullet. Find (a) the function of speed verses time of bullet after bullet enters into the sand; (b) the maximum depth that the bullet can reach.

Answer 1

Given:

A bullet with mass m is projected out horizontally into the sand with speed v0. It is subject to the resistance =k (k is a constant).

The gravity of bullet has to be neglected.

To find:

• Speed vs Time function

• Maximum depth upto which bullet can penetrate.

Calculation:

Since resistance = k is a constant , we can apply Equations of Kinematics to solve this problem :

After entering sand , let final velocity be v and time be t .

Applying 1st Equation of Kinematics :

$\therefore \: v = u + at$

$= > 0 = v_{0} + \{( - k)t \}$

$= > v_{0} = kt$

So required function is :

$\boxed{ v_{0} = kt}$

Applying 3rd Equation of Kinematics:

$\therefore \: {v}^{2} = {u}^{2} + 2as$

$= > {(0)}^{2} = {(v_{0})}^{2} + \{ 2 \times ( - k) \times s \}$

$= > {(v_{0})}^{2} = 2ks$

$= > s = \dfrac{ {(v_{0})}^{2} }{2k}$

So maximum depth reached will be :

$\boxed{ s = \dfrac{ {(v_{0})}^{2} }{2k} }$