Question

Скорость движения точки изменяется по закону V=36t-12t²Найдите путь, пройденный точкой от начала дыижения до её остановки

Answer 1

Найдем время, через которое тело остановится 
v=0
36t-12t²=0
12t(3-t)=0
t₁=0
t₂=3
Поскольку
ΔL=vΔt
то пройденный путь численно будет равен площади криволинейной трапеции
$L= \int\limits^3_0 {(36t-12t^2)} \, dt =(18t^2-4t^3)|_0^3=18*3^2-4*3^3-0=54$
Ответ: 54 ед.

Answer 2

Calculating the Distance Traveled by a Point

To find the distance traveled by a point from the beginning of its motion until it stops, we can use the equation for displacement. Given the equation for velocity as a function of time, V = 36t - 12t², we can integrate this equation to find the displacement, which will give us the distance traveled.

Calculating Displacement and Distance Traveled

The displacement (s) of the point can be found by integrating the velocity function with respect to time. The equation for displacement (s) is given by:

s(t) = ∫(V) dt

Where V is the velocity function, and the limits of integration will be from the initial time (t=0) to the time when the point stops.

Integrating the Velocity Function

Integrating the velocity function V = 36t - 12t² with respect to time will give us the displacement function s(t).

The integration of V = 36t - 12t² with respect to t yields:

s(t) = ∫(36t - 12t²) dt

This integration will provide us with the displacement function, which we can then use to find the distance traveled by the point.

Finding the Distance Traveled

Once we have the displacement function, we can find the distance traveled by the point from the beginning of its motion until it stops by taking the absolute value of the displacement function and evaluating it at the time when the point stops.

The distance traveled (d) can be calculated using the displacement function as follows:

d = |s(t_stop)|

Where t_stop is the time when the point stops.

By following these steps, we can calculate the distance traveled by the point from the beginning of its motion until it stops using the given velocity function.