A bullet is fired with an initial speed of 200 m/s. If a target is 500ft away, at what angle should the rifle be raised in order to hit the target?
To solve this problem, we can use the principles of projectile motion. The motion of a projectile can be divided into two independent components: horizontal motion (along the x-axis) and vertical motion (along the y-axis). We'll start by focusing on the vertical component.
Given:
Initial speed of the bullet (v0) = 200 m/s
Distance to the target (d) = 500 ft = 152.4 m
Let's assume that the rifle is fired horizontally, which means there is no initial vertical velocity (vy0 = 0). The only force acting on the bullet vertically is gravity.
To find the angle at which the rifle should be raised, we need to determine the time it would take for the bullet to reach the target horizontally. Once we have the time, we can then calculate the initial vertical velocity, which will allow us to find the angle using trigonometric functions.
Step 1: Finding the time of flight
Using the horizontal component of motion, we can use the formula:
Time of flight (t) = distance (d) / horizontal velocity (vx)
Since the horizontal velocity (vx) is constant and equal to the initial speed of the bullet (v0), we can substitute the given values:
t = 152.4 m / 200 m/s
t ≈ 0.762 seconds
Step 2: Finding the initial vertical velocity
Using the vertical component of motion, we can use the formula:
Vertical displacement (sy) = vertical velocity (vy0) × time (t) + (0.5 × acceleration due to gravity (g) × t²)
In this case, the vertical displacement is zero since the bullet hits the target at the same height it was fired from (sy = 0). The acceleration due to gravity (g) is approximately 9.8 m/s².
0 = vy0 × 0.762 s + (0.5 × 9.8 m/s² × (0.762 s)²
Solving for the initial vertical velocity (vy0):
vy0 = -0.5 × 9.8 m/s² × (0.762 s) ≈ -3.547 m/s
The negative sign indicates that the bullet was fired in the downward direction.
Step 3: Finding the angle
Using trigonometric functions, we can determine the angle (θ) at which the rifle should be raised:
θ = atan(vy0 / vx)
θ = atan(-3.547 m/s / 200 m/s)
θ ≈ -1.02°
Since the angle cannot be negative, we can consider its absolute value:
θ ≈ 1.02°
Therefore, the rifle should be raised by approximately 1.02 degrees above the horizontal to hit the target 500 ft away.