We know that the rate of change of displacement is called velocity and the rate of change of velocity is called acceleration. Here, instead a value for displacement and time, an equation is given. So calculate velocity and acceleration, we need to differentiate the equation.Formulas used:
V = displacement/time or v = dx/dt
and a = velocity/time or a = dv/dt = d²x/dt²
Also,
dx/dt = 1/dt/dx
TO FIND :-
the relationship between v and a = ?
SOLUTION STEP BY STEP :-
let us start with the basic definition of velocity and acceleration.
We know that the velocity v
V = displacement/time or dx/dt
where time is t and displacement is
x .Similarly, the acceleration a is defined as the rate of change of velocity a = velocity/time or
a = dv/dt = d²x/dt²
where, time is t and velocity v .
Since, t is given in terms of x. we can
differentiate t with respect to x, then we get
dt/dx = 2 ax + B Then velocity v = dx/dt = 1/dt =
1/2ax+ B then, we must differentiate
V = dx/dt = 1/2ax + B with respect t
Here, using the mathematical differentiation of x then
This may seem as a hard question at first. But this question is easy, provided you know differentiation, here we use the mathematical differentiation of xn, then derivative of x ^ n for x = n * x ^ (n - 1) here, in our sum, n = - 1 . And usina chain rule of differentiation, we get the result.Also see that dx/dt = 1/dt /dx this is the most important step in this question. Also note
that v = displacement/time or v = dx/dt
and a = velocity/time Or a = dv/dt = d²x/dt²
To calculate, a we must differentiate only v with respect to t and not dt/dx
Answers & Comments
Verified answer
UNDERSTANDING CONCEPT :-
Also,
TO FIND :-
the relationship between v and a = ?
SOLUTION STEP BY STEP :-
let us start with the basic definition of velocity and acceleration.
We know that the velocity v
V = displacement/time or dx/dt
where time is t and displacement is
x .Similarly, the acceleration a is defined as the rate of change of velocity a = velocity/time or
a = dv/dt = d²x/dt²
where, time is t and velocity v .
Since, t is given in terms of x. we can
differentiate t with respect to x, then we get
dt/dx = 2 ax + B Then velocity v = dx/dt = 1/dt =
1/2ax+ B then, we must differentiate
V = dx/dt = 1/2ax + B with respect t
Here, using the mathematical differentiation of x then
d/dx x = nx n - 1 , here, in our sum, n = -1
And using chain rule of differentiation, we get
To find the relationship between v
and a, we can substitute
v = 1/(2ax + B ) in a Then we get,
a = - 2av²
Hence the relationship between
v and a, is - 2av²
ADDITIONAL INFORMATION :-
[tex]\bf\huge Solution \: :— [/tex]
[tex]\:[/tex]
We are given the equation of motion as:
$$ t = \alpha x + \beta x^2 $$
Taking the derivative of both sides with respect to time, we get:
$$ \frac{dt}{dt} = \frac{d}{dt}(\alpha x + \beta x^2) $$
The left-hand side simplifies to 1, while the right-hand side becomes:
[tex]\frac{d}{dt}(\alpha x + \beta x^2) = \alpha \frac{dx}{dt} + 2\beta x \frac{dx}{dt}[/tex]
Now, we can express the velocity and acceleration in terms of derivatives of position with respect to time:
$$ v = \frac{dx}{dt} $$
$$ a = \frac{d^2x}{dt^2} $$
Substituting these into the equation above, we get:
$$ 1 = \alpha v + 2\beta x v $$
Simplifying this expression, we obtain the relationship between velocity and acceleration:
$$ a = \frac{dv}{dt} = \alpha \frac{d}{dt}(v) + 2\beta v \frac{dx}{dt} $$
or
[tex]\rightarrow \boxed{a = \alpha \frac{dv}{dx} + 2\beta v^2}[/tex]