Differential Equations Answers

Suppose a small cannonball weighing 98 N is shot vertically upward, with an initial velocity
v0 = 82 m/s.If air resistance is ignored and the positive direction is upward, then a model for the state of the cannonball is given by d^2s/dt^2 = −g. Since ds/dt = v(t)the last differential equation is the same as dv/dt = −g, where we take g = 9.8 m/s2.If air resistance is incorporated into the model, it stands to reason that the maximum height attained by the cannonball must be less than if air

A. If the positive direction is upward, a model for the state of the cannonball is given by
m dv/dt = −mg − kv, where m is the mass of the cannonball and k > 0 is a constant of proportionality. Suppose k = 0.0025 and find the velocity v(t) of the cannonball at time t.

B. Use the result obtained in part (a) to determine the height s(t) of the cannonball measured from ground level.

C. Find the maximum height attained by the cannonball.

In (14) of Section 1.3 we saw that a differential equation describing the velocity v of a falling mass subject to air resistance proportional to the instantaneous velocity is m dv/dt = mg - kv, where
k > 0 is a constant of proportionality. The positive direction is downward.

a. Solve the equation subject to the initial condition v(0) = v0.
vt = _____

b. Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass.

c. If the distance s, measured from the point where the mass was released above ground, is related to velocity v by ds/dt = v(t),find an explicit expression for s(t) if s(0) = 0.

A 20-volt electromotive force is applied to an LR-series circuit in which the inductance is 0.1 henry and the resistance is 50 ohms. Find the current i(t) if i(0) = 0.

i(t = _____

Determine the current as t → ∞.
lim i(t) =
t→∞

A 100-volt electromotive force is applied to an RC-series circuit in which the resistance is 500 ohms and the capacitance is 10^−4 farad. Find the charge q(t) on the capacitor if q(0) = 0.

q(t) = _____

Find the current i(t).
i(t) = ____

The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modification of (2), given in Section 3.1, is dT/dt = kS(T-Tm), where
k < 0
and Tm is a constant. Suppose that two cups A and B are filled with coffee at the same time. Initially, the temperature of the coffee is 65° C. The exposed surface area of the coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of the coffee in cup A is 38° C. If Tm = 21° C, then what is the temperature of the coffee in cup B after 30 min? (Round your answer to two decimal places.)

______ degrees Celsius

A dead body was found within a closed room of a house where the temperature was a constant 21° C. At the time of discovery the core temperature of the body was determined to be 27° C. One hour later a second measurement showed that the core temperature of the body was 25° C. Assume that the time of death corresponds to t = 0 and that the core temperature at that time was 37° C. Determine how many hours elapsed before the body was found. [Hint: Let t1 > 0 denote the time that the body was discovered.] (Round your answer to one decimal place.)

____ hr

1. A thermometer reading 20° C is placed in an oven preheated to a constant temperature. Through a glass window in the oven door, an observer records that the thermometer reads 42° C after
1/2 minute and 62° C after 1 minute. How hot is the oven?

_____ degrees Celsius

Two large containers A and B of the same size are filled with different fluids. The fluids in containers A and B are maintained at 0° C and 100° C, respectively. A small metal bar, whose initial temperature is 100° C, is lowered into container A. After 1 minute the temperature of the bar is 90° C. After 2 minutes (since being lowered into container A) the bar is removed and instantly transferred into the other container. After 1 minute in container B the temperature of the bar rises 10°. How long, measured from the start of the entire process, will it take the bar to reach 99.4° C? (Round your answer to two decimal places. Assume the final temperature being asked for is reached while the bar is container B.)

t = ___ min

1. A small metal bar, whose initial temperature was 30° C, is dropped into a large container of boiling water. How long will it take the bar to reach 80° C if it is known that its temperature increases 2° during the first second? (The boiling temperature for water is 100° C. Round your answer to one decimal place.)

____ sec

How long will it take the bar to reach 96° C? (Round your answer to one decimal place.)

_____ sec

A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t.

A(t) = ____ lb