A hunter aims directly at a target on the same level of 120m away.
a.) If the bullet leaves the gun at a speed of 250m/s, by how much will it miss the target?
b.) At what angle should the gun be aimed so the target will be hit?
(a) The bullet takes t = 120/250 = 0.48 s to reach the deer. How far will it fall in that time? About 2.26 m.
bullet fall distance = (1/2) g t^2
(b)Aim above the deer an angle A such that V*sinA*t = (g/2) t^2
sin A = (g/2)*t/V = 0.009408
A = 0.54 degrees
To answer these questions, we can use the principles of projectile motion. Projectile motion occurs when an object is launched into the air and moves along a curved path under the influence of gravity.
a.) To determine by how much the bullet will miss the target, we need to find the horizontal distance it will travel during its flight. We can use the formula:
Horizontal distance (d) = initial velocity (v) × time of flight (t)
In this case, the initial velocity is 250 m/s, and the bullet travels in a straight line. Therefore, the time of flight would be the same as the time it takes for the bullet to travel 120 m horizontally.
Time of flight (t) = distance (d) / velocity (v)
Let's now calculate the time it takes to travel 120 m horizontally:
t = 120 m / 250 m/s
t ≈ 0.48 s
Now, we can find the horizontal distance covered by the bullet:
d = 250 m/s × 0.48 s
d ≈ 120 m
Since the bullet covers a horizontal distance of 120 m, it will not miss the target horizontally. Therefore, it will not miss the target in terms of horizontal displacement.
b.) The angle at which the gun should be aimed to hit the target can be determined using the equation for the range (distance covered horizontally) of a projectile:
Range (R) = (v² * sin(2θ)) / g
Where:
- v is the initial velocity of the projectile
- θ is the angle at which the projectile is launched
- g is the acceleration due to gravity (approximately 9.8 m/s²)
In this case, we know the range (distance) is 120 m, the initial velocity is 250 m/s, and g is approximately 9.8 m/s². Therefore, we can rearrange the formula to solve for the angle (θ):
θ = (1/2) * arcsin ((R * g) / (v²))
Plugging in the given values, we get:
θ = (1/2) * arcsin ((120 m * 9.8 m/s²) / (250 m/s)²)
θ ≈ 18.86°
Therefore, the gun should be aimed at an angle of approximately 18.86° above the horizontal to hit the target 120 m away.