Question

Three displacement vectors of a croquet ball are shown in the figure below, where || = 25.0 units, || = 60.0 units, and || = 15.0 units. Find the following.

(a) The resultant displacement in unit-vector notation

resultant

=

(b) The magnitude and direction of the resultant displacement

magnitude	= units
direction	= ° (from the +x axis)

Answer 1

Given

Vector \(\vec{A}=25\) units \((j)\)

Vector \(\vec{B}=60\) units \(\cos 45(\hat{i})+60\) units \(\sin 45(\hat{j})\)

$$ \vec{B}=42.42(\hat{i})+42.42(\hat{j}) $$

Vector \(\overrightarrow{\mathrm{C}}=15\) units \(\cos 45(\hat{i})+15\) units \(\sin 45(-1)\)

$$ \overrightarrow{\mathrm{C}}=10.6(\hat{\mathrm{i}})+10.6(\hat{-j}) $$

(a) The resultant displacement.

$$ \begin{aligned} R &=(42.42+10.6)(\hat{i})+(25+42.42-10.6)(\hat{j}) \\ &=53.02(\hat{i})+56.82(\hat{j}) \end{aligned} $$

(b) The magnitude of the resultant displcem ent is

$$ \begin{aligned} R &=\sqrt{(53.02)^{2}+(56.82)^{2}} \\ &=77.7 \text { units } \end{aligned} $$

Direction, \(\theta=\tan ^{-1}\left(\frac{56.82}{53.02}\right)\)

\(\theta=47^{0}\) with respect to \(x\) axis

Answer 2

SOLUTION :

a.

Vertical displacements

= 26 + 60 sin(45) - 15 sin(45)

= 57.82 units

Horizontal displacement

= 25 cos(90) + 60 cos(45) + 15 cos(45)

= 53.03 units

Resultant displacement =  (53.03 I + 57.82 j) (ANSWER).

b.

Magnitude :

Resultant displacement (magnitude) 

= sqrt(57.82^2 - 53.03^2) 

= 78.46 units  (ANSWER).

Direction 

= atan ( 57.82/53.03) 

= 47.7º with + x axis (upward) (ANSWER).