A bird flying horizontally at 4.25 m/s
accidentally drops a rock it was carrying. A short time later, the rock's velocity is 13.0 m/s at -70.9°.
How much time has passed?
To find the time that has passed, we can use the fact that the acceleration due to gravity is 9.8 m/s².
First, we need to find the horizontal and vertical components of the rock's velocity when it was dropped.
The bird was flying horizontally, so the horizontal component of the rock's velocity remains the same, which is 4.25 m/s.
Next, we can use the vertical component of the rock's velocity to find the time it took to reach that velocity.
The vertical component of the velocity is given as -70.9° at a magnitude of 13.0 m/s, so we can use trigonometry to find the vertical component of the velocity:
Vertical component = 13.0 m/s * sin(-70.9°) = -12.493 m/s (Negative because it is downward direction)
The relationship between time, initial velocity, acceleration, and final velocity is given by the equation:
Final velocity = Initial velocity + (acceleration * time)
Using the vertical component of velocity:
-12.493 m/s = 0 m/s + (-9.8 m/s²) * time
Solving for time:
time = -12.493 m/s / (-9.8 m/s²)
time ≈ 1.27 s
Therefore, approximately 1.27 seconds have passed since the rock was dropped.
To find the time that has passed, we can use the equations of motion.
The horizontal motion (x-direction) of the rock is unaffected by the vertical motion. So, we can use the equation:
\(x = v_x \cdot t\)
where:
- \(x\) is the horizontal displacement (which is equal to 0 because the rock was dropped horizontally)
- \(v_x\) is the horizontal velocity of the rock (which is also 0 because the rock was dropped horizontally)
- \(t\) is the time that has passed
Substituting the values into the equation, we get:
\(0 = 4.25 \cdot t\)
Solving for \(t\), we find:
\(t = 0 s\)
Therefore, no time has passed since the rock was dropped.
To determine the time that has passed, we need to use the equations of motion. Let's break down the information given:
Initial velocity of the bird (horizontal component): v₀x = 4.25 m/s
Final velocity of the rock (magnitude): v = 13.0 m/s
Final velocity of the rock (angle with respect to the horizontal): θ = -70.9°
Using trigonometry, we can find the horizontal and vertical components of the final velocity of the rock:
Horizontal component of the final velocity: v_f = v * cos(θ)
Vertical component of the final velocity: u_f = v * sin(θ)
Since the bird was initially flying horizontally, the vertical component of its velocity, as well as the rock's vertical velocity component at the moment it was dropped, is zero.
Now, let's find the time it took for the rock to reach its final velocity in the horizontal direction:
The horizontal displacement (Δx) during this period is given by the equation:
Δx = v₀x * t + 0.5 * a * t^2
Where a is the horizontal acceleration, which we assume to be zero since there are no horizontal forces acting on the rock.
Simplifying the equation, we have:
Δx = v₀x * t
Using the value of v₀x (4.25 m/s) and the known horizontal displacement Δx, we can solve for time:
t = Δx / v₀x
However, we need to know the horizontal displacement (Δx) in order to calculate time. This information is not provided in the question, so it is not possible to determine the exact amount of time that has passed without additional details.