Complete Question
One day, after pulling down your window shade, you notice that sunlight is passing through a pinhole in the shade and making a small patch of light on the far wall. Having recently studied optics in your physics class, you're not too surprised to see that the patch of light seems to be a circular diffraction pattern. It appears that the central maximum is about 2 cm across, and you estimate that the distance from the window shade to the wall is about 5 m.
Required:
Estimate the diameter of the pinhole.
Answer:
The diameter is
Explanation:
From the question we are told that
The central maxima is
The distance from the window shade is
The average wavelength of the sun is mathematically evaluated as
Generally the visible light spectrum has a wavelength range between 400 nm to 700 nm
So the initial wavelength of the sun is
and the final wavelength is
Substituting this into the above equation
The diameter is evaluated as
substituting values
Answer:
4.6834625323 m/s
0 m/s
Explanation:
s = Displacement
t = Time
Velocity is given by
The bird's average velocity for the return flight is 4.6834625323 m/s
In the whole episode the bird went 5220 km away from its nest and came back. This means the displacement is zero.
Hence, the average velocity for the whole episode is 0 m/s
Answer:
His launching angle was 14.72°
Explanation:
Please, see the figure for a graphic representation of the problem.
In a parabolic movement, the velocity and displacement vectors are two-component vectors because the object moves along the horizontal and vertical axis.
The horizontal component of the velocity is constant, while the vertical component has a negative acceleration due to gravity. Then, the velocity can be written as follows:
v = (vx, vy)
where vx is the component of v in the horizontal and vy is the component of v in the vertical.
In terms of the launch angle, each component of the initial velocity can be written using the trigonometric rules of a right triangle (see attached figure):
sin angle = opposite / hypotenuse
cos angle = adjacent / hypotenuse
In our case, the side opposite the angle is the module of v0y and the side adjacent to the angle is the module of vx. The hypotenuse is the module of the initial velocity (v0). Then:
sin angle = v0y / v0 then: v0y = v0 * sin angle
In the same way for vx:
vx = v0 * cos angle
Using the equation for velocity in the x-axis we can find the equation for the horizontal position:
dx / dt = v0 * cos angle
dx = (v0 * cos angle) dt (integrating from initial position, x0, to position at time t and from t = 0 and t = t)
x - x0 = v0 t cos angle
x = x0 + v0 t cos angle
For the displacement in the y-axis, the velocity is not constant because the acceleration of the gravity:
dvy / dt = g ( separating variables and integrating from v0y and vy and from t = 0 and t)
vy -v0y = g t
vy = v0y + g t
vy = v0 * sin angle + g t
The position will be:
dy/dt = v0 * sin angle + g t
dy = v0 sin angle dt + g t dt (integrating from y = y0 and y and from t = 0 and t)
y = y0 + v0 t sin angle + 1/2 g t²
The displacement vector at a time "t" will be:
r = (x0 + v0 t cos angle, y0 + v0 t sin angle + 1/2 g t²)
If the launching and landing positions are at the same height, then the displacement vector, when the object lands, will be (see figure)
r = (x0 + v0 t cos angle, 0)
The module of this vector will be the the total displacement (65 m)
module of r =
65 m = x0 + v0 t cos angle ( x0 = 0)
65 m / v0 cos angle = t
Then, using the equation for the position in the y-axis:
y = y0 + v0 t sin angle + 1/2 g t²
0 = y0 + v0 t sin angle + 1/2 g t²
replacing t = 65 m / v0 cos angle and y0 = 0
0 = 65m (v0 sin angle / v0 cos angle) + 1/2 g (65m / v0 cos angle)²
cancelating v0:
0 = 65m (sin angle / cos angle) + 1/2 g * (65m)² / (v0² cos² angle)
-65m (sin angle / cos angle) = 1/2 g * (65m)² / (v0² cos² angle)
using g = -9.8 m/s²
-(sin angle / cos angle) * (cos² angle) = -318.5 m²/ s² / v0²
sin angle * cos angle = 318.5 m²/ s² / (36 m/s)²
(using trigonometric identity: sin x cos x = sin (2x) / 2
sin (2* angle) /2 = 0.25
sin (2* angle) = 0.49
2 * angle = 29.44
<u>angle = 14.72°</u>
