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 :

$\sf{v = u + at}$

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

$\sf{ = > v = v_{0} - kt}$

So required function is :

$\boxed{ \sf{ v = v_{0} - kt}}$

Applying 3rd Equation of Kinematics:

$\sf{ {v}^{2} = {u}^{2} + 2as}$

$\sf{ = > {(0)}^{2} = {(v_{0})}^{2} + 2as}$

$\sf{ = > s = \dfrac{ {v_{0}}^{2} }{2a} }$

So maximum depth reached will be :

$\boxed{ \sf{ s = \dfrac{ {v_{0}}^{2} }{2a} }}$