An arrow is fired vertically upwards by a bow and reaches an altitude of 134 m. Find the initial speed of the arrow on the ground level.
An arrow is fired vertically upwards by a bow and reaches an altitude of 134 m. Find the initial speed of the arrow on the ground level.
To find the initial speed of the arrow on the ground level, we can use the equations of motion for vertical motion.
The equation that relates the final velocity (Vf), initial velocity (Vi), acceleration (a), and displacement (s) is:
Vf^2 = Vi^2 + 2as
In this case, the final velocity at the highest point is 0, since the arrow reaches its maximum height before falling back down. The initial velocity is what we want to find, the acceleration is due to gravity (approximately -9.8 m/s^2), and the displacement is the altitude of 134 m.
Using the equation, we can substitute the values:
0 = Vi^2 + 2(-9.8)(134)
Simplifying the equation:
0 = Vi^2 - 2632
Rearranging the equation:
Vi^2 = 2632
Taking the square root of both sides:
Vi = ± √2632
Note that we can ignore the negative solution since speed only has positive values.
Using a calculator, the square root of 2632 is approximately 51.29.
Therefore, the initial speed of the arrow on the ground level is approximately 51.29 m/s.
To find the initial speed of the arrow on the ground level, we can use the kinematic equations of motion. In this case, we'll use the equation that relates the final velocity (when the arrow reaches its maximum height) and the initial velocity (when the arrow was fired) to the acceleration due to gravity and the displacement.
The equation we'll use is:
vf^2 = vi^2 + 2ad
Where:
vf is the final velocity (which is 0 when the arrow reaches its maximum height)
vi is the initial velocity (what we're trying to find)
a is the acceleration due to gravity (-9.8 m/s^2, assuming the arrow is fired on Earth)
d is the displacement (134 m, the height the arrow reaches)
Rearrange the equation to solve for vi:
vi^2 = vf^2 - 2ad
Given that vf = 0, the equation becomes:
vi^2 = -2ad
Now substitute the values:
vi^2 = -2 * (-9.8 m/s^2) * 134 m
Simplifying, we have:
vi^2 = 2644 m^2/s^2
To find vi, we'll take the square root of both sides:
vi = √2644 m/s
Evaluating this, we get:
vi ≈ 51.4 m/s
Therefore, the initial speed of the arrow on the ground level is approximately 51.4 m/s.