The velocity of an object is v(t) = 27 − t2, 0 ≤ t ≤ 6 where v is measured in meters per second and t is the time in seconds.

a. Find the velocity v(2) and acceleration a(2) of the object when t = 2.
b. What can be said about the speed of the object when the velocity and acceleration have opposite signs?

(a) 23 m/s, -4 m/s²

(b) Speed is decreasing.

Explanation:

The velocity of the object is given as:

v(t) = 27 - t²

When t = 2 secs, velocity, v(2) becomes:

v(2) = 27 - (2)²

v(2) = 27 - 4

v(2) = 23 m/s

The acceleration is the first derivative of the velocity, dv/dt:

a(t) = dv(t)/dt = -2t

Acceleration after 2 secs, a(2) is:

a(2) = -2*2

a(2) = -4 m/s²

(b) When velocity and acceleration have opposite signs, it means that the velocity and acceleration are in opposite directions, hence, the object is slowing down.

In other words, the speed is decreasing.

b.transmitters send radio waves, and receivers capture radio waves.

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.