Mechanics> A car is driven east for a distance of 54...

A car is driven east for a distance of 54 km, then north for 32 km, and then in a direction 28° east of north for 27 km. Draw the vector diagram and determine the total displacement of the car from its starting point.

Simran Bhatia , 9 Years ago

Grade 11

1 Answers

Aditi Chauhan

Last Activity: 9 Years ago

Assumption:

Let us assume that the vector $\overrightarrow{a}$ represents the eastward motion of the car with a magnitude of , the northward motion of the car is given by vector $\overrightarrow{b}$ with a magnitude of 32 km while the east of north motion by vector $\overrightarrow{c}$ with a magnitude of
27 km.
The figure below shows the vector diagram of the Car’s motion:

In vector $\overrightarrow{a}$ , there is no vertical component but only the horizontal component along the east with magnitude 54 km. Therefore vector $\overrightarrow{a}$ is:
$\overrightarrow{a} = 54\ km\ \widehat{i}$
For vector $\overrightarrow{b}$ , there is no horizontal component as it describes the northward motion of the car, along the unit vector $\widehat{j}$ .
The magnitude of the vertical component of the car is 32 km, therefore vector $\overrightarrow{b}$ can be written as:
$\overrightarrow{b} = 32\ km\ \widehat{j}$
From the vector diagram above, it is clear that the horizontal component of vector $\overrightarrow{c}$ points in the direction of unit vector $\widehat{i}$ . Therefore the horizontal component of vector $\overrightarrow{c}$ is .
27 km sin (28°)
The vertical component of the vector $\overrightarrow{c}$ points northward, along the unit vector $\widehat{j}$ , therefore the component is.27 km cos(28°) .
Therefore vector $\overrightarrow{c}$ is:

The magnitude of total vertical displacement (say s_v) of the car along the north is the sum of vertical component of vector $\overrightarrow{a}$ , the vertical component of vector $\overrightarrow{b}$ and the vertical component of vector $\overrightarrow{c}$ .
Since vector $\overrightarrow{a}$ does not have a vertical component, we add the vertical component of vector $\overrightarrow{b}$ and $\overrightarrow{c}$ , to obtain the magnitude of displacement s_v as:

The magnitude of total horizontal displacement (say s_h) of the car along the east is the sum of horizontal component of vector $\overrightarrow{a}$ , the horizontal component of vector $\overrightarrow{b}$ and the horizontal component of vector $\overrightarrow{c}$ .
Since vector $\overrightarrow{b}$ does not have a horizontal component, we add the horizontal component of vector $\overrightarrow{a}$ and $\overrightarrow{c}$ , to obtain the magnitude of displacement s_h as:
…
The magnitude of net displacement (say s) of the car can be calculated using the magnitude in equation (1) and (2) as:

Rounding off to two significant figures
s = 87 km
Therefore the magnitude of displacement of the car during his journey is 87 km .