A pine cone falls straight down from a pine tree growing on a 21.0^\circ slope. The pine cone hits the ground with a speed of 9.00 m/s. What is the component of the pine cone's impact velocity (a) parallel to the ground and (b) perpendicular to the ground?

To determine the component of the pine cone's impact velocity parallel to the ground and perpendicular to the ground, we can use trigonometry and break down the components of the velocity.

Given:
Slope angle (θ) = 21.0°
Impact velocity (V) = 9.00 m/s

(a) Component of velocity parallel to the ground:

The component of velocity parallel to the ground is given by:

V_parallel = V * cos(θ)

Where:
V_parallel is the component of velocity parallel to the ground.
V is the magnitude of the velocity.
θ is the slope angle.

Substituting the known values:

V_parallel = 9.00 m/s * cos(21.0°)

Calculating:

V_parallel ≈ 9.00 m/s * 0.9272

V_parallel ≈ 8.3448 m/s

Therefore, the component of the pine cone's impact velocity parallel to the ground is approximately 8.3448 m/s.

(b) Component of velocity perpendicular to the ground:

The component of velocity perpendicular to the ground is given by:

V_perpendicular = V * sin(θ)

Where:
V_perpendicular is the component of velocity perpendicular to the ground.
V is the magnitude of the velocity.
θ is the slope angle.

Substituting the known values:

V_perpendicular = 9.00 m/s * sin(21.0°)

Calculating:

V_perpendicular ≈ 9.00 m/s * 0.3894

V_perpendicular ≈ 3.5046 m/s

Therefore, the component of the pine cone's impact velocity perpendicular to the ground is approximately 3.5046 m/s.

To find the components of the pine cone's impact velocity, we need to decompose the velocity vector into its horizontal and vertical components.

Let's consider the following trigonometric relationships for a right triangle:

1. The sine of an angle is equal to the ratio of the opposite side to the hypotenuse.
2. The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse.

In this case, the slope of the ground forms an angle of 21.0 degrees with the horizontal. Therefore, the vertical component of the impact velocity (V_y) will be equal to the magnitude of the velocity (9.00 m/s) multiplied by the sine of the slope angle.

V_y = 9.00 m/s * sin(21.0^\circ)

Similarly, the horizontal component of the impact velocity (V_x) will be equal to the magnitude of the velocity (9.00 m/s) multiplied by the cosine of the slope angle.

V_x = 9.00 m/s * cos(21.0^\circ)

Now, let's calculate these values:

V_y = 9.00 m/s * sin(21.0^\circ) ≈ 3.364 m/s (a)

V_x = 9.00 m/s * cos(21.0^\circ) ≈ 8.161 m/s (b)

Therefore, the component of the pine cone's impact velocity parallel to the ground (a) is approximately 3.364 m/s, and the component perpendicular to the ground (b) is approximately 8.161 m/s.

Compute the vertical velocity from the distance that it falls, and resolv e that vector into directions along and perpendcular to the slope.

These are steps you should be able to do on your own by now.