The fastest bird is the spine-tailed swift, which reaches speeds of 171 km/h. Suppose that you wish to shoot such a bird with a 22-caliber rifle that fires a bullet with a speed of 366 m/s. If you fire at the instant when the bird is 30 m directly overhead, how many meters ahead of the bird must you aim the rifle? Ignore gravity in this problem.

To find out how many meters ahead of the bird you must aim the rifle, we need to analyze the relative velocity between the bird and the bullet.

Let's break down the velocities involved:

1. The speed of the bird: 171 km/h. To convert this to m/s, we multiply by 1000 to get 171,000 m/h, and then divide by 3600 to get 47.5 m/s.

2. The speed of the bullet: 366 m/s.

3. The vertical speed of the bird: We are told to ignore gravity in this problem, so the vertical speed of the bird can be considered zero.

Now, let's define the reference frame for this problem. Since we are ignoring gravity, we will consider the horizontal and vertical motions separately. In this case, the horizontal motion of the bird and the bullet will be independent of each other.

Since the horizontal motions are independent, we can set up an equation to find the time it takes for the bullet to reach the bird:

Distance = Speed × Time

The distance the bird travels horizontally during this time is the distance you need to aim ahead of it. Since the bird's horizontal speed is 47.5 m/s, and the bullet's speed is 366 m/s, we can write:

Distance = (47.5 m/s) × Time

Now, to find the time it takes for the bullet to reach the bird, we can use the fact that the vertical motion of the bird is zero. The time it takes for the bullet to reach the bird can be found using the formula:

Time = Distance / Relative Velocity

The relative velocity between the bird and the bullet is the difference in their horizontal speeds. So:

Relative Velocity = Bullet Speed - Bird Speed

Substituting the values:

Relative Velocity = (366 m/s) - (47.5 m/s) = 318.5 m/s

Now, we can substitute this value into our equation for time:

Time = Distance / (318.5 m/s)

We are given that the bird is directly overhead at a height of 30 meters. Therefore, the distance the bullet needs to travel horizontally is the same as the distance between the point where the bullet is fired and the bird's vertical position:

Distance = 30 meters

Finally, we can substitute this distance into the equation for time:

Time = (30 meters) / (318.5 m/s)

Calculating this value:

Time ≈ 0.094 seconds

Now that we have the time it takes for the bullet to reach the bird, we can find the distance you need to aim ahead of the bird:

Distance = (47.5 m/s) × (0.094 seconds)

Calculating this value:

Distance ≈ 4.47 meters

Therefore, you need to aim approximately 4.47 meters ahead of the bird to shoot it with your rifle.

How long does it take bullet to go 30 meters? Call that t

Then how far does the bird fly in time t?