Aiming for the Bull's Eye: Accounting for Gravity

[VashtheStampede]

an acher shatneds 40m away from his target. If the arrow is shot horizontally with a velocity of 90m/s how far above the bull's eye must he aim to compensate for gravity pulling down. No air resistance and I guess the target is level with him.

 

[berkeman]

How long does it take for the arrow to reach the target? How far would the arrow fall vertically in that amount of time?

 

[quicknote]

I would approach this problem by using the principles of projectile motion and the laws of gravity. The first step would be to break down the initial velocity of the arrow, which is 90m/s, into its horizontal and vertical components. The horizontal component of the velocity remains constant throughout the motion, while the vertical component is affected by gravity.

Since there is no air resistance, we can assume that the arrow will follow a parabolic trajectory. The horizontal distance traveled by the arrow can be calculated using the formula d = v*t, where d is the distance, v is the initial velocity, and t is the time. In this case, the horizontal distance would be 40m.

Next, we need to determine the time it takes for the arrow to reach the target. This can be done using the formula t = d/v, where t is the time, d is the distance, and v is the initial velocity. In this case, the time would be 40m/90m/s = 0.44 seconds.

Now, we can use the formula for vertical displacement in projectile motion, which is d = v*t + 0.5*a*t^2, where d is the displacement, v is the initial vertical velocity, a is the acceleration due to gravity (9.8m/s^2), and t is the time. Since the arrow is shot horizontally, the initial vertical velocity is 0. Plugging in the values, we get d = 0 + 0.5*9.8*0.44^2 = 0.97m.

Therefore, the archer must aim 0.97m above the bull's eye to compensate for gravity pulling the arrow down. It is important to note that this calculation assumes that the target is at the same level as the archer. If the target is at a different height, the calculation would be more complex, taking into account the vertical displacement of the target as well.

In conclusion, by using the principles of projectile motion and the laws of gravity, we can determine the necessary compensation for gravity when aiming at a distant target. This calculation can be useful for archers, as well as other fields such as ballistics and space exploration.