Question

Рух матеріальної точки задано рівнянням x=2t-t². Написати залежність швидкості від часу. Побудувати графік швидкості і графік прискорення.

Answer 1

Щоб знайти залежність швидкості від часу, треба взяти похідну від функції руху за час:

v = dx/dt = 2 - 2t

Отже, швидкість залежить від часу за законом v = 2 - 2t.

Для побудови графіків швидкості та прискорення необхідно знайти залежність прискорення від часу. Для цього знову беремо похідну від функції руху:

a = dv/dt = -2

Отже, прискорення є константою і дорівнює -2.

Answer 2

Equation of Motion for a Material Point

The equation of motion for a material point is given by the equation x = 2t - t², where x represents the position of the point at time t.

Deriving the Velocity Equation

To find the velocity equation, we need to differentiate the position equation with respect to time (t). Differentiating x = 2t - t² with respect to t gives us the velocity equation.

Let's differentiate x = 2t - t² with respect to t:

dx/dt = d(2t - t²)/dt

The derivative of 2t with respect to t is 2, and the derivative of -t² with respect to t is -2t. Therefore, the velocity equation is:

v = dx/dt = 2 - 2t

Graph of Velocity

To graph the velocity equation v = 2 - 2t, we can plot the velocity (v) on the y-axis and time (t) on the x-axis. The graph will be a straight line with a negative slope of -2.

Here is a graph of the velocity equation:


Deriving the Acceleration Equation

To find the acceleration equation, we need to differentiate the velocity equation with respect to time (t). Differentiating v = 2 - 2t with respect to t gives us the acceleration equation.

Let's differentiate v = 2 - 2t with respect to t:

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

The derivative of 2 with respect to t is 0, and the derivative of -2t with respect to t is -2. Therefore, the acceleration equation is:

a = dv/dt = -2

Graph of Acceleration

The acceleration equation a = -2 represents a constant acceleration of -2 units per second squared. Since the acceleration is constant, the graph of acceleration will be a horizontal line at -2 on the y-axis.

Here is a graph of the acceleration equation:


Please note that the graphs provided are for illustrative purposes only and may not accurately represent the actual graphs of the given equations.