A dart is being thrown at a board that is 2.80 m away. When it hits the board it is 0.486 m below the point where it was aimed and it was thrown at 8.90 m/s. How long was the dart in the air?
To determine the time the dart was in the air, we can use the equations of motion.
First, let's consider the vertical motion of the dart. The dart is thrown downward, so we'll consider downward as the positive direction. We have the following known values:
- Initial vertical displacement (h) = -0.486 m (negative because it is below the initial point)
- Initial vertical velocity (v₀) = 0 m/s (since the dart is already thrown at this point)
- Acceleration due to gravity (g) = 9.8 m/s² (acting downward)
We can use the following equation to find the time of flight (t):
h = v₀t + (1/2)gt²
Rearranging the equation, we get:
t = √(2h/g)
Substituting the values, we have:
t = √(2 * -0.486 m / 9.8 m/s²)
By evaluating this equation, we find that t is approximately 0.099 seconds.
Now, to calculate the horizontal distance traveled by the dart, we need to know the horizontal velocity (v) and the time of flight (t).
Given:
Horizontal distance (d) = 2.80 m
Horizontal velocity (v) = 8.90 m/s
The horizontal distance (d) can be found using the equation:
d = vt
Rearranging and substituting the values, we have:
2.80 m = 8.90 m/s * t
Solving for t:
t = 2.80 m / 8.90 m/s
Evaluating this equation, we find that t is approximately 0.315 seconds.
Therefore, the time the dart was in the air is approximately 0.315 seconds.