Question

The initial velocity u of a bullet in penetrating a distance 's' through a target is reduced by u/n.how far will the bullet proceed through the target before coming to rest?

Answer 1

Answer on Question #44306 – Physics - Mechanics | Kinematics | Dynamics

The initial velocity $u$ of a bullet in penetrating a distance 's' through a target is reduced by $u/n$. how far will the bullet proceed through the target before coming to rest?

Solution:

$u$ – initial velocity of the bullet;

$S$ – first traveled distance;

$\frac{u}{n}$ – final velocity after travelling first distance;

$D$ – traveled distance before coming to rest;

$a$ – deceleration of the bullet;

$t_1$ – time of travelling distance $S$;

$t_2$ – time of travelling distance $D$;

Equation of motion of the bullet for the travelled distance $S$:

$$S = u t_1 - \frac{a t_1^2}{2}$$

Rate equation for the bullet:

$$\frac{u}{n} = u - a t_1$$ $$a = \left(\frac{u}{t_1} - \frac{u}{n t_1}\right) = \frac{u}{t_1} \left(1 - \frac{1}{n}\right)$$

(2) in(1):

$$S = u t_1 - \frac{u}{t_1} \left(1 - \frac{1}{n}\right) \cdot \frac{t_1^2}{2}$$ $$S = u t_1 - \frac{u}{2} \left(1 - \frac{1}{n}\right) t_1$$ $$t_1 = \frac{S}{u - \frac{u}{2} \left(1 - \frac{1}{n}\right)} = \frac{S}{\frac{u}{2} + \frac{u}{2} \cdot \frac{1}{n}} = \frac{2S}{u(n + 1)}$$

(3) in(2):

$$a = \frac{u}{\frac{2S}{u(n + 1)}} \left(1 - \frac{1}{n}\right) = \frac{u^2(n + 1)}{2S} \left(1 - \frac{1}{n}\right) = \frac{u^2(n + 1)}{2S} \cdot \left(\frac{n - 1}{n}\right) = \frac{u^2(n + 1)}{2S} \cdot \left(\frac{n - 1}{n}\right) = \frac{u^2(n^2 - 1)}{2nS}$$

Rate equation for the bullet (final velocity of the bullet is zero):

$$0 = u - a t_2$$ $$t_2 = \frac{u}{a}$$

Equation of motion of the bullet for the travelled distance $D$:

$$D = u t_2 - \frac{a t_2^2}{2}$$

(5) in(6):

$$D = u \cdot \frac{u}{a} - \frac{a \left(\frac{u}{a}\right)^2}{2} = \frac{u^2}{2a}$$ $$D = \frac {u ^ {2}}{2 \cdot \frac {u ^ {2} (n ^ {2} - 1)}{2 n S}} = \frac {u ^ {2} n S}{u ^ {2} (n ^ {2} - 1)} = \frac {n S}{n ^ {2} - 1}$$

Answer: Distance that bullet proceed through the target before coming to rest is equal to $\frac{nS}{n^2 - 1}$

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