Question

Figure (3-Q1) shows the x- coordinate of a particle as a function of time. Find the signs of v_x and ax  at t=t₁, t= t₂  and t=t₃. 

Answer 1

If x' and x" be the x-coordinate of the particle at initial time t' and t" respectively then v_x  =(x"-x')/(t"-t') = tanθ. 

For t"-t' infinitesimally small it is the  v_x at that instant. 

So slope of the tangent at any point in the above graph gives v_x . 

At t=t₁, tanθ is positive, so sign of  v_x is positive. 

At t= t₂  the slope of the curve is horizontal, so tanθ=0 → v_x =0. 

At t=t₃ the slope of the curve is negative, so sign of v_x is negative.