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A rifle with a muzzle velocity of 1500 m/s shoots a bullet at a small target 150 m away. How high above the target must the rifle be aimed so that the bullet hits the target?

A. Zero
B. 4.9 cm
C. 9.8 cm
D. 98 cm

Answer :

To solve this problem, we will consider the projectile motion of the bullet. When a bullet is fired towards a target, it will follow a parabolic path due to the effect of gravity on its vertical motion. Let's analyze this step-by-step.

  1. Initial Conditions:

    • Muzzle velocity [tex]v_0 = 1500\, \text{m/s}[/tex]
    • Distance to target [tex]d = 150\, \text{m}[/tex]
    • The bullet will travel horizontally and needs to reach the target; thus, we need to calculate the time [tex]t[/tex] it takes for the bullet to travel this horizontal distance.

  2. Time to reach the target:

    • The horizontal component of the rifle's velocity is the same as the muzzle velocity since the rifle is assumed to be aimed directly at the target.
    • Using the equation for constant speed, [tex]d = v_0 \cdot t[/tex], we solve for [tex]t[/tex]:
      [tex]t = \frac{d}{v_0} = \frac{150}{1500} = 0.1\, \text{s}[/tex]

  3. Vertical Motion:

    • During the time [tex]t[/tex], gravity will cause the bullet to fall a certain distance vertically. The bullet's initial vertical velocity is zero if aimed directly at the target.
    • The vertical displacement [tex]h[/tex] due to gravity can be calculated using the equation [tex]h = \frac{1}{2} g t^2[/tex]
      where [tex]g = 9.8\, \text{m/s}^2[/tex] is the acceleration due to gravity.
    • Substituting [tex]t = 0.1\, \text{s}[/tex]:
      [tex]h = \frac{1}{2} \times 9.8 \times (0.1)^2 = 0.049\, \text{m} = 4.9\, \text{cm}[/tex]

  4. Conclusion:

    • Thus, to ensure the bullet hits the target, the rifle must be aimed [tex]4.9\, \text{cm}[/tex] above the target.

Therefore, the correct answer is (b) 4.9 cm.

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Rewritten by : Jeany

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