A particle moves in the plane so that its velocity vector at time t seconds is v(t) and its acceleration vector is a(t). At a particular instant t=t0 the dot product v(t0)⋅a(t0) is known to be positive.
Answer each of the following in complete sentences, using appropriate mathematical terminology and reasoning.
(a) Describe geometrically what the sign of v(t0)⋅a(t0) reveals about the relative directions of v(t0) and a(t0). Include a statement about the angle between the vectors.
(b) Explain, without performing any explicit calculations of speed, whether the particle’s speed is increasing or decreasing at t=t0 and justify your conclusion.
(c) State and justify what would be true about the particle’s speed if v(t0)⋅a(t0)=0, and what would be true if v(t0)⋅a(t0) were negative.