Two material points move in the same frame of reference according to the given equations.

Two material points move in the same frame of reference according to the given equations. At what point in time will the speeds of these points be the same? Find the speed and acceleration of points at this moment in time. The equation of motion of the first point, m: x = 20 + 4t-4.5t ^ 2; Equation of motion of the second point, m: x = 2 + 2t + 0.5t ^ 2

The dependence of the point’s velocity on time is the first derivative of the coordinate’s dependence on time. Then for two points we have:

v1 (t) = dx1 / dt = 4 – 9 * t;

v2 (t) = dx2 / dt = 2 + t.

At some point in time, the velocities of the points turned out to be equal, i.e., v1 = v2. Then

4 – 9 * t = 2 + t,

whence the moment in time at which the speeds are equal:

10 * t = 2;

t = 0.2 s.

Substitute this value into the formula for the speed of the second body, and find the speed of the points at this moment:

v1 = v2 = 2 + 0.2 = 2.2 m / s.

The dependence of the point acceleration on time is the first derivative of the speed dependence on time. Then for two points we have:

a1 (t) = dv1 / dt = -9;

a2 (t) = dv2 / dt = 1.

Thus, at any moment of time, the accelerations of the points are equal to -9 m / s2 and 1 m / s2, respectively.