Question

Galileo thought about whether acceleration should be defined as the rate of change of velocity over time or as the rate of change in velocity over distance. He chose the former, so let's use the name "vroomosity" for the rate of change of velocity over distance. For motion of a particle on a straight line with constant acceleration, the equation $\mathit{v}\mathbf{=}{\mathit{v}}_{\mathbf{i}}\mathbf{+}\mathit{a}\mathit{t}$gives its velocityv as a function of time. Similarly, for a particle's linear motion with constant vroomosity k, the equation $\mathit{v}\mathbf{=}{\mathit{v}}_{\mathbf{i}}\mathbf{+}\mathit{k}\mathit{x}$gives the velocity v as a function of the position xif the particle's speed is ${\mathit{v}}_{\mathbf{i}}$at x = 0. (a) Find the law describing the total force acting on this object of mass m. (b) Describe an example of such a motion or explain why it is unrealistic. Consider (c) the possibility of kpositive and (d) the possibility of knegative.

Short answer

a) The expression for the acceleration for describing the force acting on the object is a= kv .

b) As k is not in the dimension of time, so the given equation is not realistic.

c) The motion of the object with high velocities can be explained by taking positive values of k.

d) The motion of object falling in viscous fluid can be explained with negative values of k.

Step-by-step solution

Velocity and Acceleration

Velocity is defined as the rate of change of velocity with respect to time.

$v=\frac{ds}{dt}$

Here s is the displacement, t is the time and v is the velocity.

Acceleration is defined as the rate of change of velocity with respect to time.

$a=\frac{dv}{dt}$

Here a is the acceleration, t is the time and v is the velocity.

Finding the expression for the acceleration

(a)

The equation of the motion for a straight line is given by

$v=v_i+kx$....... (1)

Here v is the final velocity of the object,$v_i$is the initial velocity of the object, k is the constant of velocity and x is the position of the particle.

The acceleration of the object is given by,

$a=\frac{dv}{dt}$...... (2)

Here is the acceleration, t is the time and v is the velocity.

Substitute$v_i$$+kx$for v from equation (1) into equation (2).

$$\begin{gathered} a=\frac{d\left(v_i+kx\right)}{dt} \\ =\frac{d{v}_i}{dt}+k\frac{dx}{dt} \\ =0+kv \end{gathered}$$

So, $a=kv$...... (3)

Therefore, the expression for the acceleration for describing the force acting on the object is $a=kv$.

Explanation for (b)

(b)

If the equation (1) represents motion of the object in straight line then the term$kx$ is in the dimension of velocity and$k$ must be in the dimension of time. So the given equation is not realistic.

Explanation for (c)

(c)

The motion of the object with high velcoities with exponential growth in energy usage can be explained by the positive$\left(k>0\right)$ values of $k$.

Explanation for (d)

(d)

The motion of the object falling in viscous fluids that provides resistive force to the object can be explained with negative value of $k\left(k<0\right)$.