The concept of derivatives and integrals allows to find the results for the questions are the motion of a particle where the speed depends on time are:
a)the position is: r = 2 t³ i + t² j
b) the position of the body on the y-axis is a parabola and on the x-axis it is a cubic function
c) The acceleration is: a = 12 t i + 2 j
d) the force is: F = 60 t i + 10 j
e) the torque is: τ = 40 t³ k^
f) tha angular momentum is: L = 4t³ - 6 t² k^
g) The kinetic energy is: K = 2 m t² (9t² +1)
h) The power is: P = 2m (36 t³ + 2t)
Kinematics studies the movement of bodies, looking for relationships between position, speed and acceleration.
a) They indicate the function of speed.
v = 6 t² i + 2t j
Ask the function of the position. The velocity is defined by the variation of the position with respect to time
v = 
dr = v dt
we substitute and integrate.
∫ dr = ∫ (6 t² i + 2t j) dt
r - 0 = 
r = 2 t³ i + t² j
b) The position of the body on the y axis is a parabola and on the x axis it is a cubic function.
c) Acceleration is defined as the change in velocity with time.
a = 
a = 12 t i + 2 j
d) Newton's second law states that force is proportional to mass times the body's acceleration.
F = ma
F = m (12 t i + 2 j)
F = 5 12 t i + 2 j
F = 60 t i + 10 j
e) Torque is the vector product of the force and the distance to the origin.
τ = F x r
The easiest way to write these expressions is to solve for the determinant.
![\tau = \left[\begin{array}{ccc}i&j&k\\F_x&F_y&F_z\\x&y&z\end{array}\right]](https://tex.z-dn.net/?f=%5Ctau%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26j%26k%5C%5CF_x%26F_y%26F_z%5C%5Cx%26y%26z%5Cend%7Barray%7D%5Cright%5D)
![\tau = \left[\begin{array}{ccc}i&j&k\\60t &10&0\\2t^3 &t^2&0\end{array}\right]](https://tex.z-dn.net/?f=%5Ctau%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26j%26k%5C%5C60t%20%2610%260%5C%5C2t%5E3%20%26t%5E2%260%5Cend%7Barray%7D%5Cright%5D)
τ = (60t t² - 2t³ 10) k
τ = 40 t³ k ^
f) Angular momentum
L = r x p
L =rx (mv)
L = m (rxv)
The easiest way to write these expressions is to solve for the determinant.
![\left[\begin{array}{ccc}i&j&k\\2t^3 &t^2&0\\6t^2&2t&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26j%26k%5C%5C2t%5E3%20%26t%5E2%260%5C%5C6t%5E2%262t%260%5Cend%7Barray%7D%5Cright%5D)
L = (4t³ - 6 t²) k
g) The kinetic energy is
K = ½ m v²
K = ½ m (6 t² i + 2t j) ²
K = m 18 t⁴ + 2t²
K = 2 m t² (9t² +1)
h) Power is work per unit time
P = 
dW / dt
The relationship between work and kinetic energy
W = ΔK
P = 
p = 2m (36 t³ + 2t)
In conclusion with the concept of derivatives and integrals we can find the results for the questions are the motion of a particle where the speed depends on time are:
a) The position is: r = 2 t³ i + t² j
b) The position of the body on the y-axis is a parabola and on the x-axis it is a cubic function
c) The acceleration is: a = 12 t i + 2 j
d) The force is: F = 60 t i + 10 j
e) The torque is: τ = 40 t³ k^
f) The angular momentum is: L = 4t³ - 6 t² k^
g) The kinetic energy is: K = 2 m t² (9t² +1)
h) The power is: P = 2m (36 t³ + 2t)
Learn more here: brainly.com/question/11298125
