Question

A bullet is fired horizontally on a point target at \[100\;{\rm{cm}}\] with a speed of \[500\;{\rm{m}}{{\rm{s}}^{{\rm{ - 1}}}}\]. With what vertical distance, it will miss the target? \[\left[ {g = 10\;{\rm{m}}{{\rm{s}}^{{\rm{ - 2}}}}} \right]\]
A. \[0.04\;{\rm{mm}}\]
B. \[0.004\;{\rm{mm}}\]
C. \[0.00004\;{\rm{m}}\]
D. \[0.0045\;{\rm{m}}\]

Answer 1

Hint: The above problem can be resolved using the fundamental equations of motion, especially the second equation of motion. First of all, the time taken by the bullet to hit the target is calculated, then using the value of time obtained from this equation, one can apply the second equation of motion. In the second equation of motion, the initial speed will be zero, as the body was initially at rest. Then by making the substitution of the numerical values, one can obtain the final result.

Complete step by step answer:
Given:
The horizontal distance of the point target is, \[d = 100\;{\rm{cm}} \times \dfrac{{1\;{\rm{m}}}}{{100\;{\rm{cm}}}} = 1\;{\rm{m}}\].
The speed of the bullet is, \[v = 500\;{\rm{m}}{{\rm{s}}^{{\rm{ - 1}}}}\].
The value of gravitational acceleration is, \[g = 10\;{\rm{m/}}{{\rm{s}}^{\rm{2}}}\].
The expression for the time take to hit the ground is,
\[
v = \dfrac{d}{t}\\
t = \dfrac{d}{v}
\]
Solve by substituting the value in the above equation as,
\[
t = \dfrac{d}{v}\\
t = \dfrac{{1\;{\rm{m}}}}{{500\;{\rm{m/s}}}}\\
t = 2 \times {10^{ - 3}}\;{\rm{s}}
\]
Let the vertical distance be h, so that the bullet will miss the target.
Then the mathematical expression for the vertical distance is given by applying the second equation of motion as,
\[h = u \times t + \dfrac{1}{2}g{t^2}\]
Here, u is the initial speed of the bullet and its value is \[0\;{\rm{m/s}}\].
Solve by substituting the values as,
 \[
h = u \times t + \dfrac{1}{2}g{t^2}\\
\Rightarrow h = \left( {0\;{\rm{m/s}}} \right) \times t + \dfrac{1}{2} \times 10 \times {\left( {2 \times {{10}^{ - 3}}} \right)^2}\\
\Rightarrow h = \dfrac{1}{2} \times 10 \times { {2 \times {{10}^{ - 3}}} }\times 2 \times {{10}^{ - 3}}\\
\Rightarrow h = 0.00002\;{\rm{m}}
\]

Therefore, the vertical distance with which the bullet will miss the target is 0.00002 m and none of the options is correct.

Note:
To resolve the given problem, one must remember the fundamental equations of motions and their mathematical formulas. These formulas are useful in resolving the typical problems in kinematics, that involves the motion in 1-D, motion in 2-D and the motion in 3-D.