Question

Question: Find the components of $v_{tot}$ along the $x$- and $y$-axes in Figure 3.55.

Figure: The two velocities ${\mathrm{v}}_{\mathrm{A}}$ and ${\mathrm{v}}_{\mathrm{B}}$ add to give a total ${\mathrm{v}}_{\mathrm{t}\mathrm{o}\mathrm{t}}$.

Short answer

The $x$ component of $v_{tot}$ is $4.409\mathrm{m}/\mathrm{s}$ and $y$ component of $v_{tot}$ is $5.072\mathrm{m}/\mathrm{s}$.

Step-by-step solution

Understanding resolution of vectors

Resolution of vectors is the process of breaking a vector into two or more vectors along distinct directions, in order to ascertain the magnitude and the direction of vector components that together generate the same impact as a single vector.

Given data:

• $v_{tot}=6.72\mathrm{m}/\mathrm{s}$.

Calculating horizontal component of total velocity

Figure: the components and the resultant

The angle between$x$-axis and$v_{tot}$is

$\begin{array}{c}\alpha =\left({26.5}^{\circ }\right)+\left({22.5}^{\circ }\right)\\ ={49}^{\circ }\end{array}$

The horizontal component of$v_{tot}$is

$v_{to{t}_x}=v_{tot}\cos \alpha$

Substituting$6.72\mathrm{m}/\mathrm{s}$for$v_{tot}$and${49}^{\circ }$for$\alpha$, we have

$\begin{array}{c}{v}_{to{t}_x}=\left(6.72\mathrm{m}/\mathrm{s}\right)\times \cos \left({49}^{\circ }\right)\\ =4.409\mathrm{m}/\mathrm{s}\end{array}$

Hence, the $x$ component of $v_{tot}$ is $4.409\mathrm{m}/\mathrm{s}$.

Calculating vertical component of horizontal velocity

The vertical component of$v_{tot}$is

$v_{to{t}_y}=v_{tot}\sin \alpha$

Substituting$6.72\mathrm{m}/\mathrm{s}$for$v_{tot}$and${49}^{\circ }$for$\alpha$, we have

$\begin{array}{c}{v}_{to{t}_y}=\left(6.72\mathrm{m}/\mathrm{s}\right)\times \sin \left({49}^{\circ }\right)\\ =5.072\mathrm{m}/\mathrm{s}\end{array}$

Hence, the $y$ component of $v_{tot}$ is $5.072\mathrm{m}/\mathrm{s}$.