Solved on Mar 20, 2024

Find the $y$ -component of a velocity vector $52^{\circ}$ below the negative $x$ -axis with $x$ -component $-20 \mathrm{m} / \mathrm{s}$ , rounded to 1 decimal place.

STEP 1

Assumptions
1. The velocity vector is at an angle of $52^{\circ}$ below the negative $x$ -axis.
2. The $x$ -component of the velocity vector is $-20 \mathrm{~m/s}$ .
3. We need to find the $y$ -component of the velocity vector.
4. The angle given is with respect to the negative $x$ -axis, so we need to consider this when applying trigonometric functions.

STEP 2

The $x$ and $y$ components of the velocity vector can be found using trigonometric functions. Since the angle is given with respect to the negative $x$ -axis, we can use the cosine function for the $x$ -component and the sine function for the $y$ -component.
$v_x = v \cos(\theta)$ $v_y = v \sin(\theta)$
where $v_x$ is the $x$ -component, $v_y$ is the $y$ -component, $v$ is the magnitude of the velocity vector, and $\theta$ is the angle.

STEP 3

We are given the $x$ -component ( $v_x$ ) and need to find the $y$ -component ( $v_y$ ). We can use the relationship between the components and the angle to find the magnitude of the velocity vector ( $v$ ).
$v = \frac{v_x}{\cos(\theta)}$

STEP 4

Since the $x$ -component is negative and the angle is below the negative $x$ -axis, the $y$ -component will also be negative. This is because the vector is in the third quadrant where both $x$ and $y$ components are negative.

STEP 5

Substitute the given values into the equation to find the magnitude of the velocity vector.
$v = \frac{-20 \mathrm{~m/s}}{\cos(52^{\circ})}$

STEP 6

Calculate the magnitude of the velocity vector.
$v = \frac{-20 \mathrm{~m/s}}{\cos(52^{\circ})}$
We can use a calculator to find the cosine of $52^{\circ}$ .

STEP 7

Using a calculator, we find that $\cos(52^{\circ}) \approx 0.6157$ .

STEP 8

Now we can calculate the magnitude of the velocity vector.
$v = \frac{-20 \mathrm{~m/s}}{0.6157}$

STEP 9

Perform the division to find the magnitude of the velocity vector.
$v \approx \frac{-20 \mathrm{~m/s}}{0.6157} \approx -32.5 \mathrm{~m/s}$

STEP 10

Now that we have the magnitude of the velocity vector, we can find the $y$ -component using the sine function.
$v_y = v \sin(\theta)$

STEP 11

Substitute the values for the magnitude of the velocity vector and the angle into the equation to find the $y$ -component.
$v_y = -32.5 \mathrm{~m/s} \sin(52^{\circ})$

STEP 12

Use a calculator to find the sine of $52^{\circ}$ .

STEP 13

Using a calculator, we find that $\sin(52^{\circ}) \approx 0.7880$ .

STEP 14

Now we can calculate the $y$ -component of the velocity vector.
$v_y = -32.5 \mathrm{~m/s} \times 0.7880$

STEP 15

Perform the multiplication to find the $y$ -component of the velocity vector.
$v_y \approx -32.5 \mathrm{~m/s} \times 0.7880 \approx -25.6 \mathrm{~m/s}$
The value of the $y$ -component of the velocity vector is approximately $-25.6 \mathrm{~m/s}$ , to one decimal place.

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Find the $y$ -component of a velocity vector $52^{\circ}$ below the negative $x$ -axis with $x$ -component $-20 \mathrm{m} / \mathrm{s}$ , rounded to 1 decimal place.

STEP 1

STEP 2

STEP 3

We are given the $x$ -component ( $v_x$ ) and need to find the $y$ -component ( $v_y$ ). We can use the relationship between the components and the angle to find the magnitude of the velocity vector ( $v$ ).
$v = \frac{v_x}{\cos(\theta)}$

STEP 4

Since the $x$ -component is negative and the angle is below the negative $x$ -axis, the $y$ -component will also be negative. This is because the vector is in the third quadrant where both $x$ and $y$ components are negative.

STEP 5

Substitute the given values into the equation to find the magnitude of the velocity vector.
$v = \frac{-20 \mathrm{~m/s}}{\cos(52^{\circ})}$

STEP 6

Calculate the magnitude of the velocity vector.
$v = \frac{-20 \mathrm{~m/s}}{\cos(52^{\circ})}$
We can use a calculator to find the cosine of $52^{\circ}$ .

STEP 7

Using a calculator, we find that $\cos(52^{\circ}) \approx 0.6157$ .

STEP 8

Now we can calculate the magnitude of the velocity vector.
$v = \frac{-20 \mathrm{~m/s}}{0.6157}$

STEP 9

Perform the division to find the magnitude of the velocity vector.
$v \approx \frac{-20 \mathrm{~m/s}}{0.6157} \approx -32.5 \mathrm{~m/s}$

STEP 10

Now that we have the magnitude of the velocity vector, we can find the $y$ -component using the sine function.
$v_y = v \sin(\theta)$

STEP 11

Substitute the values for the magnitude of the velocity vector and the angle into the equation to find the $y$ -component.
$v_y = -32.5 \mathrm{~m/s} \sin(52^{\circ})$

STEP 12

Use a calculator to find the sine of $52^{\circ}$ .

STEP 13

Using a calculator, we find that $\sin(52^{\circ}) \approx 0.7880$ .

STEP 14

Now we can calculate the $y$ -component of the velocity vector.
$v_y = -32.5 \mathrm{~m/s} \times 0.7880$

STEP 15

Perform the multiplication to find the $y$ -component of the velocity vector.
$v_y \approx -32.5 \mathrm{~m/s} \times 0.7880 \approx -25.6 \mathrm{~m/s}$
The value of the $y$ -component of the velocity vector is approximately $-25.6 \mathrm{~m/s}$ , to one decimal place.

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Find the yyy-component of a velocity vector 52∘52^{\circ}52∘ below the negative xxx-axis with xxx-component −20m/s-20 \mathrm{m} / \mathrm{s}−20m/s, rounded to 1 decimal place.

STEP 1

STEP 2

STEP 3

We are given the xxx-component (vxv_xvx​) and need to find the yyy-component (vyv_yvy​). We can use the relationship between the components and the angle to find the magnitude of the velocity vector (vvv).v=vxcos⁡(θ)v = \frac{v_x}{\cos(\theta)}v=cos(θ)vx​​

STEP 4

Since the xxx-component is negative and the angle is below the negative xxx-axis, the yyy-component will also be negative. This is because the vector is in the third quadrant where both xxx and yyy components are negative.

STEP 5

Substitute the given values into the equation to find the magnitude of the velocity vector.v=−20 m/scos⁡(52∘)v = \frac{-20 \mathrm{~m/s}}{\cos(52^{\circ})}v=cos(52∘)−20 m/s​

STEP 6

Calculate the magnitude of the velocity vector.v=−20 m/scos⁡(52∘)v = \frac{-20 \mathrm{~m/s}}{\cos(52^{\circ})}v=cos(52∘)−20 m/s​We can use a calculator to find the cosine of 52∘52^{\circ}52∘.

STEP 7

Using a calculator, we find that cos⁡(52∘)≈0.6157\cos(52^{\circ}) \approx 0.6157cos(52∘)≈0.6157.

STEP 8

Now we can calculate the magnitude of the velocity vector.v=−20 m/s0.6157v = \frac{-20 \mathrm{~m/s}}{0.6157}v=0.6157−20 m/s​

STEP 9

Perform the division to find the magnitude of the velocity vector.v≈−20 m/s0.6157≈−32.5 m/sv \approx \frac{-20 \mathrm{~m/s}}{0.6157} \approx -32.5 \mathrm{~m/s}v≈0.6157−20 m/s​≈−32.5 m/s

STEP 10

Now that we have the magnitude of the velocity vector, we can find the yyy-component using the sine function.vy=vsin⁡(θ)v_y = v \sin(\theta)vy​=vsin(θ)

STEP 11

Substitute the values for the magnitude of the velocity vector and the angle into the equation to find the yyy-component.vy=−32.5 m/ssin⁡(52∘)v_y = -32.5 \mathrm{~m/s} \sin(52^{\circ})vy​=−32.5 m/ssin(52∘)

STEP 12

Use a calculator to find the sine of 52∘52^{\circ}52∘.

STEP 13

Using a calculator, we find that sin⁡(52∘)≈0.7880\sin(52^{\circ}) \approx 0.7880sin(52∘)≈0.7880.

STEP 14

Now we can calculate the yyy-component of the velocity vector.vy=−32.5 m/s×0.7880v_y = -32.5 \mathrm{~m/s} \times 0.7880vy​=−32.5 m/s×0.7880

STEP 15

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Find the $y$ -component of a velocity vector $52^{\circ}$ below the negative $x$ -axis with $x$ -component $-20 \mathrm{m} / \mathrm{s}$ , rounded to 1 decimal place.

We are given the $x$ -component ( $v_x$ ) and need to find the $y$ -component ( $v_y$ ). We can use the relationship between the components and the angle to find the magnitude of the velocity vector ( $v$ ).
$v = \frac{v_x}{\cos(\theta)}$

Since the $x$ -component is negative and the angle is below the negative $x$ -axis, the $y$ -component will also be negative. This is because the vector is in the third quadrant where both $x$ and $y$ components are negative.

Substitute the given values into the equation to find the magnitude of the velocity vector.
$v = \frac{-20 \mathrm{~m/s}}{\cos(52^{\circ})}$

Calculate the magnitude of the velocity vector.
$v = \frac{-20 \mathrm{~m/s}}{\cos(52^{\circ})}$
We can use a calculator to find the cosine of $52^{\circ}$ .

Using a calculator, we find that $\cos(52^{\circ}) \approx 0.6157$ .

Now we can calculate the magnitude of the velocity vector.
$v = \frac{-20 \mathrm{~m/s}}{0.6157}$

Perform the division to find the magnitude of the velocity vector.
$v \approx \frac{-20 \mathrm{~m/s}}{0.6157} \approx -32.5 \mathrm{~m/s}$

Now that we have the magnitude of the velocity vector, we can find the $y$ -component using the sine function.
$v_y = v \sin(\theta)$

Substitute the values for the magnitude of the velocity vector and the angle into the equation to find the $y$ -component.
$v_y = -32.5 \mathrm{~m/s} \sin(52^{\circ})$

Use a calculator to find the sine of $52^{\circ}$ .

Using a calculator, we find that $\sin(52^{\circ}) \approx 0.7880$ .

Now we can calculate the $y$ -component of the velocity vector.
$v_y = -32.5 \mathrm{~m/s} \times 0.7880$