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?
(Unit = s)
(Hint: the rock was originally moving with the bird.)
We can analyze the horizontal and vertical components of the rock's motion separately.
Given that the bird was flying horizontally at 4.25 m/s, the horizontal velocity of the rock when it was dropped is also 4.25 m/s.
Let's focus on the vertical component of motion. The vertical velocity of the rock when it was dropped is 0 m/s because it was moving horizontally. The final vertical velocity of the rock is given as -13.0 m/s (negative because it is moving downwards).
Using the kinematic equation, v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time elapsed, we can find the time elapsed for the vertical motion.
Final vertical velocity (v) = initial vertical velocity (u) + acceleration (a) * time (t)
-13.0 m/s = 0 m/s + (-9.8 m/s^2) * t
Solving for t:
-13.0 m/s = -9.8 m/s^2 * t
t = -13.0 m/s / -9.8 m/s^2
t ≈ 1.33 s
Since the rock was originally moving with the bird, the time elapsed for the rock is the same as the time elapsed for the bird. Therefore, approximately 1.33 seconds have passed.
To find the time that has passed, we can use the equation for horizontal velocity:
v = Δx / t
where v represents the horizontal velocity, Δx represents the change in horizontal distance, and t represents time.
Since the horizontal velocity remains constant at 4.25 m/s, Δx = 0. Therefore, the bird and the rock didn't cover any horizontal distance during the time interval we are interested in.
The vertical velocity of the rock changes from an unknown value to 13.0 m/s at an angle of -70.9°. From this information, we can determine the change in vertical velocity (Δv) as follows:
Δv = v_f - v_i
where v_f is the final vertical velocity (13.0 m/s) and v_i is the initial vertical velocity.
We are given the magnitude and direction of the final velocity, but we need to determine the vertical component of the initial velocity. To do this, we can use trigonometry.
Using the given angle (-70.9°), we can find the vertical component of the initial velocity (v_iy). The horizontal component (v_ix) remains constant at 4.25 m/s.
v_iy = v_i * sin(angle)
Since the rock was originally moving with the bird, both bird and rock had the same initial vertical velocity.
Therefore, v_iy = 4.25 * sin(-70.9°) = -3.766 m/s
Now, we can calculate the change in vertical velocity (Δv):
Δv = v_f - v_iy
Δv = 13.0 - (-3.766) = 16.766 m/s
Now we can find the time (t) using the equation:
Δv = a * t
where a is the acceleration due to gravity, which is approximately 9.8 m/s².
t = Δv / a
t = 16.766 / 9.8 ≈ 1.710 seconds
Therefore, approximately 1.710 seconds have passed.
To find the time that has passed, we need to use the kinematic equation for velocity with the given information.
The kinematic equation for velocity is:
vf = vi + at,
Where:
- vf is the final velocity,
- vi is the initial velocity,
- a is the acceleration,
- t is the time.
Given:
- The bird's velocity, which is also the rock's initial velocity: vi = 4.25 m/s,
- The rock's final velocity: vf = 13.0 m/s,
- The angle at which the rock moves: -70.9°.
Since the bird was flying horizontally and the rock was moving at an angle, we need to separate the velocities into their x and y components. The horizontal component of the rock's velocity will be equal to the bird's velocity.
The x-component of the rock's final velocity (vrxf) can be found using the formula:
vrxf = vix
Since the bird was flying horizontally and dropping the rock vertically, the y-component of the rock's final velocity (vryf) can be found using the formula:
vryf = viy + ayt,
Where:
- ayt is the vertical acceleration of the rock.
Since the rock is dropped, the vertical acceleration is due to gravity acting on it, and its value is 9.8 m/s^2 (assuming no air resistance).
Now, we have two equations:
vrxf = vix (Equation 1)
vryf = viy + ayt (Equation 2)
Since vrxf = 13.0 m/s and vix = 4.25 m/s, we can write:
13.0 m/s = 4.25 m/s (Equation 3)
From Equation 2, we can see that the vertical component of the rock's initial velocity (viy) is zero because the bird was flying horizontally. Therefore, Equation 2 becomes:
vryf = 0 + (9.8 m/s^2) * t
13.0 m/s = 0 + (9.8 m/s^2) * t (Equation 4)
Now, solve Equation 3 for Equation 4:
4.25 m/s = (9.8 m/s^2) * t
Solving for t, we get:
t = 4.25 m/s / (9.8 m/s^2)
t ≈ 0.433 s
Therefore, approximately 0.433 seconds have passed since the rock was dropped.