A particle moves on a straight line?

Question

A particle moves on a straight line so that the velocity at time t, is v=4s, where s is the distance from the origin. If s=3, when t=0 , then, when t=0.5 , s=?

Is this parametrics?

Answer 1

v=ds/dt =4s so 1/4 ds/s=dt and 1/4 lnIsI =t+c
lnIsI^1/4 =t+c and s =Ke^4t At t=0 s =K=3

s= 3 e^4t When t=0.5 s= 3e^2=22.17

Answer 2

this is a simple differential equation - ie use integration.

v = dx/dt = 4x

thus 1/x dx = 4dt

ln x = 4t + c

when t=0, x=3 so c = ln 3

ln x = 4t + ln 3
ln x - ln 3 = 4t
ln (x/3) = 4t
x/3 = e^4t

x = 3e^4t

solved.

Answer 3

you should see s as a function of t
thus : v(t) = 4s(t)

you have to calculate the distance at time is T

since v'(t) = s(t)
and v'(t) = 4s'(t)
it follows that s(t) = 4s'(t)

solving this gives 1 = 4s'/s =>s(T) - 3 = 4log(T) - 4log(3)

thus s(T) = 3 + 4log(T) - 4log(3)
so when T=0.5, s = ... use calculator.