Velocity and Position Question?

let v=velocity of particle

since v is proportional to x, we can write this way: v=kx, where k is the constant of proportionality.

from the Definition of velocity(rate of change of displacement with time)

we can deduce that v=dx/dt=kx. this implies dx/x= kdt, after intergrating this equation we get lnx=kt+c

substituting the values of t and x at the start in the equation, this will give ln(2)=k(0)+c: hence c=ln2

now equation looks like this: lnx=kt+ln2, we may substitute the corresponding values of t and x to find value of k!

ln(4)=k(10)+ln2, but 4 can be written as 2^2,

so ln2^2=k(10)+ln2

2ln2=10k+ln2

k=ln2/10

now our equation looks likes this: lnx=(ln2/10)t + ln2

to answer the question, we just substitute the value of x in the above equation which will give us the result as x=2.828

so you have dx / dt = k x which means x(t) = C e^(kt), x(0) = 2 implies C = 2, x(10) = 4 means 4 = 2 e^(10k) = 2 [ e^k]^10 ---> e^k = (2)^(1/10)---> x = 2*(2^(t/10))---> x(5) = 2* √2 = 2 √2