Lesson Using vectors to solve problems in Mechanics: Velocity

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<H2>Using vectors to solve problems in Mechanics: Velocity</H2>

<H3>Problem 1</H3><TABLE>
 <TR>
 <TD>A rowboat moves across the river. &nbsp;A rower in the boat heads a direction&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

perpendicular to the river's banks &nbsp;(Figure 1a). &nbsp;The boat's speed in this
direction is 2 m/s. &nbsp;The river current speed is 1 m/s.

Draw the vector of the full velocity of the boat relative to the river's bank.

You can assume that the river's banks are parallel straight lines.

Solution

The full vector of the boat velocity relative the river's banks has two components.

From one side, &nbsp;the boat moves across the river with oars having the velocity

component &nbsp;2&nbsp; m/s &nbsp;perpendicular to the river's banks due to rowing.

From the other side, &nbsp;the boat moves along the river having the velocity component

1&nbsp; m/s &nbsp;in this direction relative to the river banks due to current.

</TD>
 <TD>
{{{drawing( 272, 210, 1.0, 10.1, -1.0, 6.0,
 
 line ( 2, 0, 9, 0), 
 line ( 2, 5, 9, 5),

blue(line ( 3,    4,   4,    4)),
            line ( 4,    4,   3.7,  4.1),
            line ( 4,    4,   3.7,  3.9),

blue(line ( 3,    3,   4,    3)),
            line ( 4,    3,   3.7,  3.1),
            line ( 4,    3,   3.7,  2.9),

blue(line ( 3,    2,   4,    2)),
            line ( 4,    2,   3.7,  2.1),
            line ( 4,    2,   3.7,  1.9),

blue(line ( 3,    1,   4,    1)),
            line ( 4,    1,   3.7,  1.1),
            line ( 4,    1,   3.7,  0.9),

line ( 5.85,  1,   5.85,  1.8),
            line ( 6.15,  1,   6.15,  1.8),

line ( 5.85,  1,   6.0,   0.8), 
            line ( 6.15,  1,   6.0,   0.8),

line ( 5.85,  1.8, 6.0,   2.0), 
            line ( 6.15,  1.8, 6.0,   2.0),

line ( 5.85,  1.5,  6.15,  1.5),
            circle (5.98, 1.45, 0.13,  0.13),

line ( 5.9,   1.4,  5.3,   1.2),
            line ( 6.1,   1.4,  6.7,   1.2),

line ( 6.85,  2,   6.85,  2.8),
            line ( 7.15,  2,   7.15,  2.8),

line ( 6.85,  2,   7.0,   1.8), 
            line ( 7.15,  2,   7.0,   1.8),

line ( 6.85,  2.8, 7.0,   3.0), 
            line ( 7.15,  2.8, 7.0,   3.0),

line ( 6.85,  2.5,  7.15,  2.5),
            circle (6.98, 2.45, 0.13,  0.13),

line ( 6.9,   2.4,  6.3,   2.2),
            line ( 7.1,   2.4,  7.7,   2.2)

)}}}

Figure 1a. &nbsp;The river, the current 
and the rowboat crossing the river&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</TD>
 <TD>
{{{drawing( 272, 210, 1.0, 10.1, -1.0, 6.0,
 
 line ( 2, 0, 9, 0), 
 line ( 2, 5, 9, 5),

blue(line ( 4,    1,   4,    3)), 
             line( 4,    3,   3.9,  2.7),
             line( 4,    3,   4.1,  2.7),

blue(line ( 4,    3,   5,    3)), 
             line( 5,    3,   4.7,  3.1),
             line( 5,    3,   4.7,  2.9),

red(line ( 4,    1,   5,    3)),
             line( 5,    3,   4.7,  2.7),
             line( 5,    3,   4.85, 2.50),

arc ( 4,    1,   1.2, 1.2, 270, 296),
             arc ( 4,    1,   1.4, 1.4, 270, 296),
           locate (4.05, 2.2, alpha),

locate (2.5, 2.6, 2(m/s)),
           locate (3.6, 4.3, 1(m/s)),

locate (4.8, 2.6, 2.236(m/s))

)}}}

Figure 1b. &nbsp;The full vector of the 
boat velocity relative to the river banks

</TD>
 </TR>
</TABLE>

The &nbsp;Figure 1b represents the full vector of the boat velocity relative to the river's banks. &nbsp;It has the component &nbsp;2&nbsp; m/s in the direction perpendicular to the banks and the component &nbsp;1&nbsp; m/s in the direction along to the river. &nbsp;The angle {{{alpha}}} of the &nbsp;Figure 1b&nbsp; is &nbsp;{{{tan(alpha)}}} = {{{1/2}}}, &nbsp;so &nbsp;{{{alpha}}} = {{{atan(1/2)}}} = 0.4636 rad = 26.65�.

The full speed &nbsp;(the magnitude of the velocity relative to the river banks)&nbsp; is &nbsp;{{{sqrt(2^2+1^2)}}} = {{{sqrt(5)}}} = 2.236 m/s.

Answer. The full vector of the boat velocity has the components &nbsp;2&nbsp; m/s &nbsp;in the direction perpendicular to the river bank, &nbsp;and &nbsp;1&nbsp; m/s &nbsp;in the direction along the current. &nbsp;This vector makes the angle &nbsp;26.65�&nbsp; with the straight line perpendicular to the river banks. &nbsp;The magnitude of this vector &nbsp;(the full speed of the boat relative to the river banks)&nbsp; is &nbsp;2.236&nbsp; m/s.

<H3>Problem 2</H3><TABLE>
 <TR>
 <TD>A river whose width is &nbsp;200 m, &nbsp;has the current speed of &nbsp;1 m/s.

A rowboat is crossing the river. &nbsp;A rower heads due a direction perpendicular&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

to the river's banks rowing at the speed of &nbsp;2 m/s &nbsp;relative to the water.

At what distance the rowboat will get the other bank counting downstream

from the point &nbsp;O&nbsp; opposed to the starting point &nbsp;D &nbsp;(Figure 2a)?

You can assume that the river's banks are parallel straight lines.

Solution

The rowboat has the speed of &nbsp;2 m/s &nbsp;in the direction perpendicular to the banks

due to rowing. &nbsp;Hence, &nbsp;it will take &nbsp;{{{(200 m)/(2(m/s))}}} = 100 seconds &nbsp;to cross the river

and to get the opposite bank.

</TD>
 <TD>
{{{drawing( 272, 210, 1.0, 10.1, -1.0, 6.0,
 
 line ( 2, 0, 9.8, 0), 
 red(circle( 5.0, 0.0, 0.15)), 
 locate( 5.1, 0.6, D),
 locate( 4.0, -0.15, Departure_point),

circle( 5.0,  5.0, 0.15), 
           locate( 5.1,  4.9, O),

line ( 2,    5,   9.8,    5), 
       red(circle( 8.5,  5.0, 0.15)), 
           locate( 8.5,  4.9, A),
           locate( 6.8,  5.7, Arrival_point),