How to do these questions?

[luke_rulz94]

Hey guys,

I'm having a bit of trouble doing these questions.
Can anyone help ?


An object moves in a line, so that its acceleration, a m/s2, at time t seconds,t > 0 is given by a = 10 e^-t/5.
a   If the object starts from rest at the origin find expressions for
i   the velocity, v m/s, in terms of t
ii   the displacement, x m from the origin, in terms of t.
b   Hence, by expressing a in terms of v, find the terminal velocity of the object.

3   A particle is projected vertically upwards against air resistance and the acceleration at time t seconds is given by x(two dots on top of the x) = -g - 1/120 v, where v m/s is the velocity. If the initial velocity is 20g m/s, prove that the object takes 20 loge 2 seconds to reach the highest point.

[Jenny_2108]

1. a) You antidifferentiate acceleration => -50(e^-t/5)+c
        The object starts from the rest, at t=0, v=0 => c=50
         v=50-50e^-t/5
   
    b) Antidifferentiate velocity => x=250(e^-t/5)+50t+d
       The object starts from the origin, at t=0, x=0 => d=-250
       x=250(e^-t/5)+50t-250

    c) From (a): (50-v)/50=e^-t/5
        Given: a=10e^-x/5=10(50-v)/50=(50-v)/5
        I've never heard "terminal velocity" before or is it in Physics? Anyways, I guess v approaches 50m/s

3. a=dv/dt=-g- 1/120 v => dt/dv=120/(-120g-v)
You antidifferentiate to get t in term of v
Sub t=0, v=20 to find c
To reach the highest point v=0, sub v=0 into equation of t => you will get the answer

3. If the initial velocity is 20g m/s, prove that the object takes 20 loge 2 seconds to reach the highest point.

Why is the initial velocity 20g m/s? Did you type wrong?