The 1D velocity of an animated space alien is well-represented by the expression v(t)= t3 -11t2 +38t -40 with t in seconds and v in m/sec. Assume that x(1)=0 m and consider the motion between t = 1 and t = 6 sec. (a) What are the (implied) units of the number 11 in the expression for v(t)? (b) During what times is the alien moving in the positive direction? At what times is the alien at rest? (c) What is the average acceleration of the alien between 2 and 5 sec? (d) What is the instantaneous acceleration of the alien at t = 5 sec? (e) What is the average velocity of the alien between 1 and 5 sec (f) Where is the alien at t = 4 sec? (x = ?)

5. Equations of the form y’ = P(x)*y^2 + Q(x)*y + R(x) are called Riccati equations.

i) If we know a solution y = φ(x) of this equation, then any other solution

can be written in the form y(x) = φ(x)+ 1/v(x), where v(x) is an unknown

function which satisfies a certain linear equation. Using the fact that

φ and y both solve the above Riccati equation, find the differential

equation that v satisfies.

ii) Consider the equation 3y’ + y^2 +2/(x^2) = 0. Find one solution of this equation by inspection.

iii) Use the method of part(i) to find the general solution of the equation

in (ii).

A skydiver, weighing 180 lb (including equipment) falls vertically downward from an altitude of 4000 ft and opens the parachute after 10 s of free fall. Assume that the force of air resistance, which is directed opposite to the velocity, is of magnitude 0.75|v| when the parachute is closed and is of magnitude 10|v| when the parachute is open, where the velocity v is measured in ft/s. (A computer algebra system is recommended. Use g = 32 ft/s2 for the acceleration due to gravity. Round your answers to two decimal places.)

Find the distance fallen before the parachute opens.

What is the limiting velocity v_Lafter the parachute opens?

Determine how long the sky diver is in the air after the parachute opens.

The capacitors referred to in this problem have only empty space between the plates.

(a) Show that a parallel-plate capacitor has a displacement current in the region between its plates that is given by Id = C dV/dt, where C is the capacitance and V is the potential difference between the plates. (Do this on paper. Your instructor may ask you to turn in this work.)

(b) A 5.00-nF parallel-plate capacitor is connected to an ideal ac generator so the potential difference between the plates is given by V = V0 cos ωt, where V0 = 3.00 V and ω = 500π rad/s. Find the displacement current in the region between the plates as a function of time. (Use the following as necessary: t and π.) Id = _____________µA

A ball of unknown mass m is tossed straight up with initial speed v. At the moment it is released, the ball is a height h above a spring-mounted platform, as shown in the figure below. The ball rises, peaks, and falls back toward the platform, ultimately compressing the spring a maximum distance d from its relaxed position. Assume that the spring is perfectly ideal with spring constant k, and that the mass of the spring and platform is negligible.

What is the mass m, assuming that there is no friction or air resistance? Using g to represent the acceleration due to gravity, enter an expression for m in terms of g, h, d, k, and v.

Using g to represent the acceleration due to gravity, enter the expression for m in terms of g, h, d, k, and v.

5. Equations of the form y’ = P(x)*y^2 + Q(x)*y + R(x) are called Riccati equations.

i) If we know a solution y = φ(x) of this equation, then any other solution

can be written in the form y(x) = φ(x)+ 1/v(x), where v(x) is an unknown

function which satisfies a certain linear equation. Using the fact that

φ and y both solve the above Riccati equation, find the differential

equation that v satisfies.

ii) Consider the equation 3y’ + y^2 +2/(x^2) = 0. Find one solution of this equation by inspection.

iii) Use the method of part(i) to find the general solution of the equation

in (ii).

A student wishes to determine the chloride ion concentration in a water sample at 25 °C using a galvanic cell constructed with a graphite electrode and a half-cell of AgCl(s) + e⁻ → Ag(s) + Cl⁻(aq) E°red = 0.2223 V And a copper electrode with 0.500 M Cu²⁺ as the second half cell Cu²⁺(aq) + 2 e⁻ → Cu(s) E°red= 0.337 V The measured cell potential when the water sample was placed into the silver side of the cell was 0.0657 V.

What is the standard potential of the cell?

What is the value of standard free energy (in kJ) for this reaction?

What is the balanced equation for the overall reaction in acidic solution?

The measured cell potential is 0.0657, what is the concentration of chloride ions in the solution?

A three phase engine has an usable power of 90 kW and an efficiency of 91%. The rating plate indicates the following values:
- 90 kW
- Power factor: 0.89 (i)
- Voltage (volts): 230 delta - 400 wye
Taking as a reference of line voltages V1=V∠90º; direct sequence and a phase oltage of the supply network of 230 V, find:
a) The phasors of the line and phase voltages of the supply network
b) The true, reactive and apparent power
c) The phasors of the line and phase currents
When the engine is connected to a supply network of 400 V and direct sequence, find:
d) The phasors of the line and phase voltages of the supply network
e) The true, reactive and apparent powers
f) The phasors of the line and phase currents

A parallel plate capacitor with plate separation d is connected to a battery. The capacitor is fully charged to Q Coulombs and a voltage of V. (C is the capacitance and U is the stored energy.) Answer the following questions regarding the capacitor charged by a battery. For each statement below, select True or False.

After being disconnected from the battery, inserting a dielectric with κ will increase U.
With the capacitor connected to the battery, decreasing d increases U.
With the capacitor connected to the battery, inserting a dielectric with κ will increase C.
With the capacitor connected to the battery, increasing d increases C.
After being disconnected from the battery, increasing d decreases V.
After being disconnected from the battery, inserting a dielectric with κ will decrease V.

Consider a modification to the DFS procedure such that it returns for an undirected graph
G, the connected components of G. To do this, you will modify the DFS procedure to
• maintain a counter i, initially set in 1, that counts how many connected components we found so
far; and
• on each vertex v ∈ G.V maintain an attribute component indicating to which connected component
number v belong to. That is, when DFS terminates, v.component indicates the component number
1, 2, . . . i that v belongs to.
Answer the following questions:
(a) Write the pseudocode for this slightly modified version of the DFS algorithm
(b) Explain why your algorithm is correct and its running time. You do not need to provide a formal
argument for correctness, but be sure to explain in clear terms.