Vectors vectors vectors?

Question

Please explain what you are doing and use calculus NOT PHYSICS! Thanks yall!


An object initially (t=0) at point (0,0,0) and having an initial velocity of <8,0,0> m/s experiences a time-dependent force that gives it an acceleration of <0,4,2t> m/s^2.

a) Find the speed of this object at t=1s.

b) How far has the object travelled in the first second?

c) Find the position of the object at t=1s.

d) Find the tangential and normal components of the acceleration at t=1s.

Answer 1

acceleration = 0*i+4*j+2t*k
velocity is the integral=0*i+4t*j+t^2*k +V where V is a constant vector
at t=0 velocity = 8*i =V
so velocity=8*i+4t*j+t^2k
distance is the integral of velocity
d= 8t*i+2t^2*j+t^3/3*k +D
d(0)= (0,0,0) so D=0
d=8*t+2t^2*j+t^3/3*k
a)at t=1 s the velocity vector is(8,4,1) and the speed is its modulus =sqrt(64+16+1)=9m/s
c) d(1) = (8,2,1/3)
b) Int (0,1) of sqrt( dx^2+dy^2+dz2) = Int(64+16t^2+t^4)^1/2dt
d) at t=1 the acceleration is (0,4,2).You have to fin d the proyection of it on the velocity vector

Answer 2

relies upon on the textbook author. There are 2 distinctive techniques of defining "components". One may well be the two perpendicular vectors that upload as much as make the vector. the different may well be the scalar values (numbers) you multiply via unit vectors in x and y to make those 2 perpendicular vectors. If the unit vector in x is stated as x0 and the unit vector in y is stated as y0, then a vector v = a*x0 + b*y0. you ought to the two call "a*x0" the x-factor, or you ought to call a the x-factor. the two one is functional, and distinctive authors will do it in yet otherwise. a number of those issues are stated as "conventions", and each textbook will inform you what convention they are adopting whilst they introduce the thought.