Force and Motion: A 12 kg armadillo runs onto a large pond of level, frictionless ice. The armadillo's initial velocity is 5m/s along the positive direction of the x-axis. Take its initial position on the ice as being the origin. It slips over the ice while being pushed by a wind with a force of 17N in the positive direction of the y-axis. In unit vector notation, what are the animal's (a) velocity and (b) position vector when it has slid for 3 seconds?

Question

Force and Motion: A 12 kg armadillo runs onto a large pond of level, frictionless ice. The armadillo's initial velocity is 5m/s along the positive direction of the x-axis. Take its initial position on the ice as being the origin. It slips over the ice while being pushed by a wind with a force of 17N in the positive direction of the y-axis. In unit vector notation, what are the animal's (a) velocity and (b) position vector when it has slid for 3 seconds?

Answer 1

There is no force in the x direction so the x velocity, call it u, remains constant at 5 m/s.

There is an acceleration, 17/12 m/s^2 in the y direction
The initial y velocity, call it Vi, is 0
so
v = Vi + a t
v = 0 + (17/12)(3)

the velocity vector at 3 seconds is therefore
V = 5 i + (17/4) j

the position in x is 5 *3 = 15 meters
the position in y is
y = (1/2) a t^2 = (1/2)(17/12)(9)
so
position vector = 15 i +(17*3/8)j

Answer 2

To find the animal's velocity and position vector, we can break down the problem into two separate components: motion in the x-axis and motion in the y-axis.

(a) Let's start by finding the animal's velocity vector. The force from the wind acting in the y-axis direction will not affect the animal's velocity in the x-axis. Therefore, only the initial velocity will contribute to the x-component of the velocity vector. Since there is no acceleration or any other forces acting on the animal in the x-axis, the x-component of the velocity will remain constant.

Given:
Initial velocity in the x-axis (Vx) = 5 m/s

To write the velocity vector in unit vector notation, we can use the x and y-components of the velocity vector:
Velocity vector (V) = Vx * î + Vy * ĵ

Since there is no change in velocity in the y-axis and there is no initial velocity in the y-axis, the y-component of the velocity can be zero.

Therefore, the velocity vector (V) = 5 m/s * î + 0 * ĵ = 5 m/s * î.

(b) Now, let's find the position vector of the animal after sliding for 3 seconds. We need to consider both the x-axis and y-axis components separately.

Given:
Initial position on the ice (x0) = 0 m
Time (t) = 3 s

For the x-axis, since there is no acceleration, the position can be found using the formula:
x = x0 + Vx * t

Plugging in the values:
x = 0 + 5 m/s * 3 s
x = 15 m

For the y-axis, we need to consider the force from the wind. The force results in acceleration according to Newton's second law, F = ma. However, since the ice is frictionless, there is no friction force or any opposing forces acting in the y-axis. Therefore, the force from the wind will directly cause an acceleration in the y-axis.

The mass (m) of the armadillo plays a role in determining the acceleration (a) in the y-axis: F = ma. From the given information, the force (F) is 17 N, so we need to determine the acceleration using the mass.

Given:
Mass (m) = 12 kg
Force (F) = 17 N

Using F = ma, we can solve for acceleration:
17 N = 12 kg * a

Solving for a:
a = 17 N / 12 kg
a ≈ 1.42 m/s²

Now, we can calculate the displacement in the y-axis using the formula:
y = y0 + Vyi * t + (1/2) * a * t²

Since there is no initial position in the y-axis and no initial velocity in the y-axis:
y = 0 + 0 * 3 s + (1/2) * 1.42 m/s² * (3 s)²
y = (1/2) * 1.42 m/s² * 9 s²
y ≈ 6.39 m

To write the position vector in unit vector notation, we can use the x and y-components of the position vector:
Position vector (r) = x * î + y * ĵ

Plugging in the values:
Position vector (r) = 15 m * î + 6.39 m * ĵ

Therefore, the animal's (a) velocity vector is 5 m/s * î, and (b) position vector after sliding for 3 seconds is 15 m * î + 6.39 m * ĵ.