What is the acceleration?

The figure shows the position of a car (black circles) at one-second intervals. What is the acceleration at the time T = 4 s?

Solution:
t=T-1s
x=v₀t+at²/2
40m=v₀∙4s+a∙(4s)²/2
13m=v₀∙1s+a∙(1s)²/2=(v₀+a∙1s/2)∙1s
v₀=13m/s-a∙1s/2
40m=(13m/s-a∙1s/2)∙4s+a∙(4s²)/2 = 52m-2s²∙a+8s²∙a = 52m+6s²∙a
6s²∙a=-12m
a=-2m/s²

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Physics Problem. Check again algebraic transformations

A rocket, speeding along toward Alpha Centauri, has an acceleration 
a(t) = At². 
Assume that the rocket began at rest at the Earth (x = 0) at t = 0. Assuming it simply travels in a straight line from Earth to Alpha Centauri (and beyond), what is the ratio of the speed of the rocket when it has covered half the distance to the star to its speed when it has travelled half the time necessary to reach Alpha Centauri?
Solution:
T - total time necessary to reach Alpha Centauri
D - distance to the star
v - speed
v₁=v(D/2)
v₂=v(T/2)
v(t)=∫At²dt=⅓At³
x(t)=∫v(t)dt=⅓At³dt=⅓¼At⁴
D=AT⁴/12
D/2=AT⁴/24
D/2=At⁴/12; At⁴/12=AT⁴/24; t⁴=T⁴/2; t⁴/T⁴=½; t/T=∜½
v₁/v₂ = v(t)/v(T/2) = {⅓At³} / {⅓A(T/2)³}
=t³ / (T/2)³ =2³ (t/T)³ = 2³ ⨯ (∜½)³ = (2∜½)³ = (∜16⨯∜½)³ = (∜8)³=4∜2
This result is not among the proposed answers to choose from. Check again algebraic transformations.

Magnetic Field and Magnetic Forces: Problems

Chapter 20, problem 31.

A rectangular x₁ cm by x₂ cm circuit carrying an x₃ A current is oriented with its plane parallel to a uniform x₄ T magnetic field (Figure 20.62, page 693).
(a) Find the magnitude and direction of the magnetic force on each straight segment of this rectangular circuit.
Illustrate your answers with clear diagrams.
(b) Find the magnitude of the net force on the entire circuit.
(c)  Find the magnitude of the net torque on the entire circuit.

Chapter 20, problem 57.

A circular metal loop is x₁ cm in diameter.
How large a current must flow through this metal so that the magnetic field at its center is equal to the magnetic field of x₂ T?

Chapter 20, problem 62.

Two circular concentric loops of wire lie on a tabletop, one inside the other.
The inner loop has a diameter of x₁ cm and carries a clockwise current of x₂ A,
as viewed from above, and the outer wire has a diameter of x₃ cm.
What must be the magnitude and direction (as viewed from above) of the current in the outer loop so that the net magnetic field due to this combination of loops is zero at the common center of the loops?

Chapter 20, problem 64.

A solenoid contain N coils of very thin wire evenly wrapped over a length of x₁ cm.
Each coil is x₂ cm in diameter.
If yhis solenoid carries a current of x₃ A, what is the magnetic field at its center?