An electron's position is given by r = 2.00t i - 7.00t2 j + 5.00 k, with t in seconds and r in meters.
(a) In unit-vector notation, what is the electron's velocity v(t)? (Answer in terms of i, j, k, and t.)
(b) What is v in unit-vector notation at t = 2.00 s?
Ive tried about 50 different "unit-vector notations" and none have been working.
The vector velocity is dr/dt, where r is the position vector.
V = 2 i - 14 t j
There is no k component.
Plug in t = 2 for part )b)
To find the electron's velocity in unit-vector notation, we need to differentiate the position equation with respect to time t. Let's start with part (a) and find the velocity vector v(t).
(a) The position vector is given by r = 2.00t i - 7.00t^2 j + 5.00 k
To find the velocity vector v(t), we differentiate each component of r with respect to t:
v(t) = dr/dt = (d(2.00t)/dt) i + (d(-7.00t^2)/dt) j + (d(5.00)/dt) k
Simplifying and taking the derivatives:
v(t) = 2.00 i - 14.00t j + 0 k
So, the electron's velocity in unit-vector notation is v(t) = 2.00 i - 14.00t j.
Now, let's move on to part (b) and find v at t = 2.00 s.
(b) To find v at t = 2.00 s, we substitute t = 2.00 into the velocity equation:
v(2.00) = 2.00 i - 14.00(2.00) j
Simplifying:
v(2.00) = 2.00 i - 28.00 j
Therefore, the electron's velocity at t = 2.00 s, in unit-vector notation, is v = 2.00 i - 28.00 j.