A hunter aims directly at a target (on the same level) 120 m away. (a) If the bullet leaves the gun at a speed of 290 m/s, by how much will it miss the target?
(b) At what angle should the gun be aimed so the target will be hit?
distance = rate x time
120 m = 290 m/s x time
time = ??
The gravitational force will be acting on the bullet during that time. The distance it will fall is
S = 1/2 (g)t^2
I get something like 0.8 m or about 80 cm.
Check my thinking. Check my work.
To solve this problem, we can use principles of projectile motion. Let's break it down into two parts:
(a) To find how much the bullet will miss the target, we need to determine the horizontal distance traveled by the bullet. We know that the bullet's initial velocity is 290 m/s and the time it takes to hit the target is the same as the time it takes to fall a vertical distance of 120 m (since the target is at the same level).
First, we'll calculate the time it takes for the bullet to hit the target. We can use the equation of motion for vertical displacement:
y = v0y * t + (1/2) * a * t^2
Since the bullet is fired horizontally with an initial vertical velocity of 0 m/s, the equation simplifies to:
120 = 0 * t + (1/2) * (-9.8) * t^2
Now we can solve for t:
120 = -4.9 * t^2
t^2 = 120 / 4.9
t ≈ 5.35 s
Now, we can calculate the horizontal distance traveled by the bullet:
x = v0x * t
Since the initial horizontal velocity is 290 m/s:
x = 290 * 5.35
x ≈ 1550.5 m
Therefore, the bullet will miss the target by approximately 1550.5 m.
(b) To find the angle at which the gun should be aimed to hit the target, we can use the concept of the vertical and horizontal components of velocity.
The initial velocity of the bullet can be resolved into horizontal (v0x) and vertical (v0y) components:
v0x = v0 * cos(theta)
v0y = v0 * sin(theta)
Where theta is the angle of the gun with respect to the horizontal.
We already know the horizontal distance traveled by the bullet is 120 m and the time of flight is approximately 5.35 s. Using these values, we can set up the following equation:
120 = v0x * t
120 = v0 * cos(theta) * t
Now we'll solve for theta:
theta = arccos(120 / (v0 * t))
Substituting the given values:
theta = arccos(120 / (290 * 5.35))
Calculating this using a calculator, the angle theta comes out to approximately 9.6 degrees.
Therefore, the gun should be aimed at an angle of approximately 9.6 degrees above the horizontal to hit the target.