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Response Details:
The position function is:
r( t ) = < 4 cos t, √2 sin t >
r( π / 4 )
= < 4 cos ( π / 4 ), √2 sin ( π / 4 ) >
= < 4 ( √2 / 2 ), √2 ( √2 / 2 ) >
= < 2√2, 1 >
The velocity function is:
v( t )
= r'( t )
= < -4 sin t, √2 cos t >
v( π / 4 )
= < -4 sin ( π / 4 ), √2 cos ( π / 4 ) >
= < -4 ( √2 / 2 ), √2 ( √2 / 2 ) >
= < -2√2, 1 >
The speed function is:
| v( t ) |
= sqrt( ( -4 sin t )2 + ( √2 cos t )2 )
= sqrt( 16 sin2 t + 2 cos2 t )
| v( π / 4 ) |
= sqrt( 16 sin2 ( π / 4 ) + 2 cos2 ( π / 4 ) )
= sqrt( 16 ( 1 / 2 ) + 2 ( 1 / 2 ) )
= sqrt( 8 + 1 )
= sqrt( 9 )
= 3
The acceleration function is:
a( t )
= v'( t )
= < -4 cos t, -√2 sin t >
a( π / 4 )
= < -4 cos ( π / 4 ), -√2 sin ( π / 4 ) >
= < -4 ( √2 / 2 ), -√2 ( √2 / 2 ) >
= < -2√2, -1 >
| a( t ) |
= sqrt( ( -4 cos t )2 + ( -√2 sin t )2 )
= sqrt( 16 cos2 t + 2 sin2 t )
| a( π / 4 ) |
= sqrt( 16 cos2 ( π / 4 ) + 2 sin2 ( π / 4 ) )
= sqrt( 16 ( 1 / 2 ) + 2 ( 1 / 2 ) )
= sqrt( 8 + 1 )
= sqrt( 9 )
= 3
Find the angle between the velocity and acceleration vectors when t = π / 4 as follows:
v( π / 4 ) dot a( π / 4 )
= < -2√2, 1 > dot < -2√2, -1 >
= ( -2√2 )( -2√2 ) + ( 1 )( -1 )
= 8 - 1
= 7
v( π / 4 ) dot a( π / 4 ) = | v( π / 4 ) | | a( π / 4 ) | cos θ =>
7 = ( 3 )( 3 ) cos θ =>
cos θ = 7 / 9 =>
θ = cos-1 ( 7 / 9 ) =>
θ = 0.679673819 rad =>
θ = 38.9424413 deg
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