Question

Could you please post a detailed answer for both parts I am having trouble understanding one of the solutions I found in the book.

1.45** Prove that if v(t) is any vector that depends on time (for example the velocity of a moving particle) but which has constant magnitude, then v(t) is orthogonal to v(t). Prove the converse that if v(t) is orthogonal to v(t), then [v(t) is constant. (Hint: Consider the derivative of v2.] This is a very handy result. It explains why, in two-dimensional polars, df/dt has to be in the direction of O and vice versa. It also shows that the speed of a charged particle in a magnetic field is constant, since the acceleration is perpendicular to the velocity.

Answer 1

solution Let ut) = Vo sin utitto coswtj so it is time dependent velocity vector but the magnitude of the velocity vector in always vo AO VE) = delta and d VW) = Von cosate Vow simati OR Últ)" = vo w [cos włê - sin dt ] and vit) = vo [simatît coset}] Also (t) = (t) = V0²wl sinut coswt (bot Product o) ?= vo [o] As dot Product as uity and its is equal to zero bo Ý It is orthogonal to ult) Also I vitsat = Vow [ sinwtit coswt 1 OR. DU = Volsinut it coswt f] so if It) in orthogonal to ū (+). then V (t) = vo[sinutit cooot f] u time dependent velocity with Constant magnitude - Sinatronat UW