Respuesta :
Answer
given,
x = 4 t³ - 6 t² + 12
velocity, [tex]v = \dfrac{dx}{dt}[/tex]
[tex]\dfrac{dx}{dt}=\dfrac{d}{dt}(4t^3-6t^2+12)[/tex]
[tex]v =12t^2-12t[/tex]
For minimum velocity calculation we have differentiate it and put it equal to zero.
[tex]\dfrac{dv}{dt} =\dfrac{d}{dt}12t^2-12t[/tex]
[tex]\dfrac{dv}{dt} =24t-12[/tex].........(1)
putting it equal to zero
24 t - 12 =0
t = 0.5 s
At t = 0.5 s velocity will be minimum.
b) minimum velocity
v = 12t² -12 t
v = 12 x 0.5² -12 x 0.5
v = -3 m/s
c) derivative of velocity w.r.t. time is acceleration
from equation 1
a = 24 t - 12
time at which acceleration will be zero
0 = 24 t - 12
t = 0.5 s
At t = 0.5 s acceleration will be zero.
Part A. The particle reaches its minimum velocity at 0.5 seconds.
Part B. The minimum velocity of the particle is -3 m/s.
Part C. The acceleration of the particle will be zero at the time t = 0.5 seconds.
How do you calculate the minimum velocity and acceleration?
Given that the position of a particle is given by the function x.
[tex]f(x) = 4t^2 -6t^2 +12[/tex]
The function of the velocity of a particle can be obtained by the time function.
[tex]v= \dfrac {dx}{dt}[/tex]
[tex]v = \dfrac {d}{dt} ( 4t^3-6t^2 +12)[/tex]
[tex]v = 12t^2 -12 t[/tex]
The velocity function of the particle is [tex]v = 12t^2 - 12t[/tex].
Part A
The minimum velocity of the particle is obtained by the differentiation of velocity function with respect to the time and put it equal to zero.
[tex]\dfrac {dv}{dt} = \dfrac {d}{dt} (12t^2 - 12t) = 0[/tex]
[tex]\dfrac {dv}{dt} = 24 t-12 = 0[/tex]
[tex]t = 0.5\;\rm s[/tex]
Hence we can conclude that the particle reaches its minimum velocity at 0.5 seconds.
Part B
The velocity function is [tex]v = 12t^2 - 12t[/tex]. Substituting the value of t = 0.5 s to calculate the minimum velocity.
[tex]v = 12(0.5)^2 - 12(0.5)[/tex]
[tex]v = 3 - 6[/tex]
[tex]v = -3 \;\rm m/s[/tex]
The minimum velocity of the particle is -3 m/s.
Part C
The acceleration is defined as the change in the velocity with respect to time. Hence,
[tex]a = \dfrac {dv}{dt}[/tex]
[tex]a = 24 t-12[/tex]
Substituting the value of a = 0, we get the time.
[tex]0 = 24t - 12[/tex]
[tex]t = 0.5 \;\rm s[/tex]
The acceleration of the particle will be zero at the time t = 0.5 seconds.
To know more about acceleration and velocity, follow the link given below.
https://brainly.com/question/2239252.
