A duck flies south at a constant speed of 20 m/s. A man stands 30 meters below the duck, he fires a shot upward at an initial velocity of 30 m/s. Half a second later the duck instantly commences a constant acceleration forward of 16 meters/sec^2. How far north should the duck be at the moment the shot is fired, if the shot is to hit the duck?
To solve this problem, we need to calculate the distance the duck travels north during the time it takes for the shot to reach its height. Let's break down the problem step by step:
Step 1: Calculate the time it takes for the shot to reach its maximum height:
The initial vertical velocity of the shot is 30 m/s, which is opposite to the gravitational acceleration of -9.8 m/s². The equation to calculate the time it takes for an object to reach its maximum height is:
t = (Vf - Vi) / a
Where:
t = time
Vf = final velocity (0 m/s when the shot reaches maximum height)
Vi = initial velocity (30 m/s)
a = acceleration (-9.8 m/s²)
Plugging in the given values:
t = (0 - 30) / -9.8
Simplifying the equation:
t = 30 / 9.8
Calculating the time:
t ≈ 3.061 seconds
Step 2: Calculate the distance the duck travels north during the time it takes for the shot to reach its maximum height.
The duck is flying south at a constant speed of 20 m/s for 3.061 seconds. Since distance equals speed multiplied by time, we can calculate the distance the duck travels during this time:
Distance = Speed × Time
Distance = 20 m/s × 3.061 s
Calculating the distance:
Distance ≈ 61.22 meters
Therefore, the duck should be approximately 61.22 meters north at the moment the shot is fired for the shot to hit the duck.