A 'displacement' always consists of a magnitude and a direction. The two cars you just described have displacements with the same magnitude ... 5 km. But if they didn't both drive in the same direction, then their displacements are different.
Remember:
-- 10 m/s² up and 10 m/s² down are different accelerations
-- 30 mph East and 30 mph West are the same speed but different velocity.
-- 5 km North and 5 km South are the same distance but different displacement.
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1st derivative gives velocity;
d r(t)/ dt = 2t i + 6 j + 4/t k
2nd derivative gives acceleration;
d^2 r(t)/ dt^2 = 2 i - 4/ t^2
Speed ;
Square root of (4 t^2 + 36 + 16/ t^2)
For a given time, like 2 seconds, t will be 2. And answer of speed will be scalar.
-- reduce the length of a wire to 1/2 . . . cut the resistance in half
-- reduce the diameter to 1/4 . . . reduce the cross-section area by (1/4²) . . . increase the resistance by 16x .
-- R2 = (R1) · (1/2) · (16) = 8 · R1
<em>-- R2 / R1 = 8</em>
