A rifle that shoots bullets at 404 m/s is to be aimed at a target 49.7 m away. If the center of the target is level with the rifle, how high (in cm) above the target must the rifle barrel be pointed so that the bullet hits dead center?

Question

A rifle that shoots bullets at 404 m/s is to be aimed at a target 49.7 m away. If the center of the target is level with the rifle, how high (in cm) above the target must the rifle barrel be pointed so that the bullet hits dead center?

Answer 1

The range is

R = v^2 sin2θ / g

so,
404^2 sin2θ /9.8 = 49.7
sin2θ = 9.8*49.7/404^2 = .00298
θ = 0.085°

not much of an elevation

Answer 2

To determine how high the rifle barrel must be pointed, we need to consider the horizontal and vertical components of the bullet's motion.

Let's start by finding the time it takes for the bullet to reach the target. We can use the equation of motion:

distance = velocity × time

Rearranging the equation, we have:

time = distance / velocity

Substituting the given values, we have:

time = 49.7 m / 404 m/s ≈ 0.123 s

Now, let's calculate the vertical displacement (height above the target) using the equation of motion for vertical motion:

displacement = initial velocity × time + (1/2) × acceleration × time^2

Since the bullet starts and lands at the same vertical position (level with the target), the initial and final displacements are both zero. Therefore, the equation simplifies to:

0 = 0 + (1/2) × acceleration × time^2

Solving for acceleration, we get:

acceleration = - (2 × displacement) / time^2

Since the bullet needs to hit the target dead center, the displacement is half the height we're looking for. Let's call this value 'h'. Substituting the values, we have:

acceleration = - (2 × (0.5h)) / (0.123 s)^2

Simplifying further, we have:

acceleration = - 16.26h

Now, let's consider the vertical component of the bullet's initial velocity. The initial vertical velocity can be calculated using the equation:

vertical velocity = acceleration × time

Substituting the values, we have:

vertical velocity = -16.26h × 0.123 s

Finally, let's convert this vertical velocity to centimeters per second for convenience.

1 m/s = 100 cm/s, so:

vertical velocity = -16.26h × 0.123 s × 100 cm/s

Simplifying, we get:

vertical velocity = -20h

Therefore, the rifle barrel must be pointed 20 times the height above the target in centimeters to hit the target dead center.