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.
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.
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
Answer:
1 second later the vehicle's velocity will be:
5 seconds later the vehicle's velocity will be:
Explanation:
Recall the formula for the velocity of an object under constant accelerated motion (with acceleration ""):
Therefore, in this case and
so we can estimate the velocity of the vehicle at different times just by replacing the requested "t" in the expression:
Answer: 288.8 m
Explanation:
We have the following data:
is the time it takes to the child to reach the bottom of the slope
is the initial velocity (the child started from rest)
is the angle of the slope
is the length of the slope
Now, the Force exerted on the sled along the ramp is:
(1)
Where is the mass of the sled and
its acceleration
In addition, if we draw a free body diagram of this sled, the force along the ramp will be:
(2)
Where is the acceleration due gravity
Then:
(3)
Finding :
(4)
(5)
(6)
Now, we will use the following kinematic equations to find :
(7)
(8)
Where is the final velocity
Finding from (7):
(9)
(10)
Substituting (10) in (8):
(11)
Finding :