Question

A vehicle moves along a straight road. The vehicle's position is given by f(t), where t is measured in seconds since the vehicle starts moving. During the first 10 seconds of the motion, the vehicle's acceleration is proportional to the cube root of the time since the start. Which of the following differential equations describes this relationship, where k is a positive constant?

a. df/dt= k3√t
b. df/dt= k3√f
c. d^2f/dt^2= k3√t
d. d^2f/dt^2= k3√f

Answer 1

Answer:

C. $\frac{d^{2}f}{dt^{2}} = k\cdot \sqrt[3] {t}$

Step-by-step explanation:

Physically speaking, the acceleration is equal to the second derivative of the position of the vehicle in time. After a careful reading to the statement, we construct the following ordinary differential equation:

$\frac{d^{2}f}{dt^{2}} = k\cdot \sqrt[3] {t}$

Where:

$t$ - Time, measured in seconds.

$k$ - Proportionality constant, measured in meters per second up to 7/3.

Therefore, the correct answer is C.

Answer 2

The differential equation that models the situation is:

$\frac{d^2f}{dt^2} = k\sqrt[3]{t}$

Which is option C.

The position is the integral of the acceleration, with is the integral of the acceleration, hence, the relation between the position and the acceleration is:

$\frac{d^2f}{dt^2}$

The acceleration is proportional to the cube root of the time since the start, hence:

$\frac{d^2f}{dt^2} = k\sqrt[3]{t}$

In which k is the constant of proportionality.

A similar problem is given at https://brainly.com/question/14480252