Taylor is interested in archery. She draws her bow and shoots an arrow at angle of 50°. It goes a distance of 100 meters. At what other angle could Taylor shoot an arrow with the same amount of force and have it go the same distance?
To find the other angle at which Taylor could shoot an arrow with the same distance, we can use the concept of projectile motion. The path of a projectile can be split into two perpendicular components: horizontal and vertical.
Given that Taylor shoots the arrow at an angle of 50°, the vertical component of the arrow's initial velocity can be calculated using the formula:
Vertical component = Initial velocity * sin(angle)
The horizontal component of the arrow's initial velocity can be calculated using the formula:
Horizontal component = Initial velocity * cos(angle)
Since the distance traveled is the same in both cases, the horizontal component of the velocity will remain constant.
Now, to find the other angle at which the arrow will travel the same distance, we need to consider the horizontal component of the velocity. The horizontal component of the velocity will be the same as in the previous case. Let's call it Vx.
So, using the formula above:
Vx = Initial velocity * cos(angle)
We can rearrange the formula to solve for the initial velocity:
Initial velocity = Vx / cos(angle)
Now, we need to find the new angle by rearranging the formula for the vertical component of the velocity:
Vertical component = Initial velocity * sin(angle)
Rearranging, we have:
sin(angle) = Vertical component / Initial velocity
Sin(angle) = Vertical component / (Vx / cos(angle))
Simplifying further, we have:
sin(angle) = Vertical component * cos(angle) / Vx
To find the new angle, we can now use the inverse sine function:
angle = inverse sin(Vertical component * cos(angle) / Vx)
Substituting the given values, where the vertical component is the same as before and Vx is the value we calculated earlier, we can solve for the new angle:
angle = inverse sin(Vertical component * cos(50°) / Vx)
Now, we can plug in the values and calculate the new angle.