Question

Материальная точка движется прямолинейно по закону x(t)= -1/3t^4+4t^3-7t^2-5t-5(где x — расстояние от точки отсчета в метрах, t — время в секундах, измеренное с начала движения). Найдите ее скорость (в м/с) в момент времени t=6 с.

Answer 1

Скорость - это первая производная функции координаты Х(t),
найдем первую производную
v(t) = x'(t) = (-1/3t⁴ + 4t³-7t² -5t -5)' = -1/3 * 4t³+4*3t²-7*2t-5*1-0=
=-4/3t³+12t²-14t-5

теперь. чтобы найти скорость в момент времени t=6с , подставим это значение вместо t
x'(6) = v(6) = -4/3*6³+12*6²-14*6-5 =-288+432-84-5 = 55 (м/с)
ответ: 55 м/с 

Answer 2

Given Information

We are given the equation for the position of a material point as a function of time:

x(t) = -1/3t^4 + 4t^3 - 7t^2 - 5t - 5

Where: - x is the distance from the reference point in meters. - t is the time in seconds, measured from the start of the motion.

We need to find the velocity of the material point at the moment t = 6 seconds.

Finding the Velocity

To find the velocity of the material point, we need to take the derivative of the position function with respect to time.

Let's differentiate the position function x(t) with respect to t:

v(t) = dx(t)/dt

Differentiating each term of the position function, we get:

v(t) = d/dt (-1/3t^4 + 4t^3 - 7t^2 - 5t - 5)

Taking the derivative of each term, we have:

v(t) = -4/3t^3 + 12t^2 - 14t - 5

Now, we can substitute t = 6 seconds into the velocity function to find the velocity at that moment:

v(6) = -4/3(6)^3 + 12(6)^2 - 14(6) - 5

Calculating this expression, we get:

v(6) = -288 + 432 - 84 - 5

v(6) = 55 m/s

Therefore, the velocity of the material point at t = 6 seconds is 55 m/s.