An archer shoots an arrow horizontally at a target 13m away. The arrow is aimed directly at the center of the target, but it hits 57cm lower. What is the initial velocity?
To calculate the initial velocity of the arrow, we can use the kinematic equations of motion. Specifically, we can use the equation that relates the horizontal displacement, initial velocity, time, and acceleration.
In this case, the horizontal displacement is given as 13m, and the vertical displacement (the arrow hitting 57cm lower) can be considered negligible, as it does not affect the horizontal motion of the arrow. Therefore, we can assume that the vertical displacement is zero.
The equation we will use is:
Displacement (horizontal) = initial velocity × time
Since the vertical displacement is zero and the arrow is shot horizontally, the time taken for the arrow to cover the horizontal distance will be the same as if the arrow were falling freely under gravity.
Using the equation for vertical displacement under constant acceleration due to gravity:
Displacement (vertical) = 0.5 × acceleration × time^2,
where the vertical displacement is -57cm (negative because it is below the target), acceleration is -9.8 m/s^2 (taking into account gravity), and time is the same in both equations.
Plugging in the values:
-0.57m = 0.5 × (-9.8m/s^2) × time^2
Rearranging the equation to solve for time:
-0.57m = -4.9m/s^2 × time^2
time^2 = 0.57m / 4.9m/s^2
time = √(0.57m / 4.9m/s^2)
time ≈ 0.3409s
Since the horizontal displacement is given as 13m, we can calculate the initial velocity. Using the formula:
Displacement (horizontal) = initial velocity × time
13m = initial velocity × 0.3409s
initial velocity = 13m / 0.3409s
initial velocity ≈ 38.13 m/s
Therefore, the initial velocity of the arrow is approximately 38.13 m/s.