Page 87 - Schaum's Outline of Differential Equations

70 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 7

7.46. A body of unknown temperature is placed in a refrigerator at a constant temperature of 0° F If after 20 minutes the
temperature of the body is 40° F and after 40 minutes the temperature of the body is 20° F, find the initial temper-
ature of the body.
7.47. A body at a temperature of 50° F is placed in an oven whose temperature is kept at 150° F. If after 10 minutes the
temperature of the body is 75° F, find the time required for the body to reach a temperature of 100° F.
7.48. A hot pie that was cooked at a constant temperature of 325° F is taken directly from an oven and placed outdoors
in the shade to cool on a day when the air temperature in the shade is 85° F. After 5 minutes in the shade, the
temperature of the pie had been reduced to 250° F. Determine (a) the temperature of the pie after 20 minutes and
(b) the time required for the pie to reach 275° F.
7.49. A cup of tea is prepared in a preheated cup with hot water so that the temperature of both the cup and the brewing
tea is initially 190° F. The cup is then left to cool in a room kept at a constant 72° F. Two minutes later, the temper-
ature of the tea is 150° F. Determine (a) the temperature of the tea after 5 minutes and (b) the time required for the
tea to reach 100° F.

7.50. A bar of iron, previously heated to 1200° C, is cooled in a large bath of water maintained at a constant temperature
of 50° C. The bar cools by 200° in the first minute. How much longer will it take to cool a second 200°?

7.51. A body of mass 3 slugs is dropped from a height of 500 ft in a with zero velocity. Assuming no air resistance, find
(a) an expression for the velocity of the body at any time t and (b) an expression for the position of the body at any
time t with respect to the coordinate system described in Fig. 7-5.
7.52. (a) Determine the time required for the body described in the previous problem to hit the ground, (b) How long
would it take if instead the mass of the body was 10 slugs?
7.53. A body is dropped from a height of 300 ft with an initial velocity of 30 ft/sec. Assuming no air resistance, find (a)
an expression for the velocity of the body at any time t and (b) the time required for the body to hit the ground.
7.54. A body of mass 2 slugs is dropped from a height of 450 ft with an initial velocity of 10 ft/sec. Assuming no air
resistance, find (a) an expression for the velocity of the body at any time t and (b) the time required for the body to
hit the ground.

7.55. A body is propelled straight up with an initial velocity of 500 ft/sec in a vacuum with no air resistance. How long
will it take the body to return to the ground?

7.56. A ball is propelled straight up with an initial velocity of 250 ft/sec in a vacuum with no air resistance. How high
will it go?

7.57. A body of mass m is thrown vertically into the air with an initial velocity v 0. The body encounters no air resistance.
Find (a) the equation of motion in the coordinate system of Fig. 7-6, (b) an expression for the velocity of the body
at any time t, (c) the time t m at which the body reaches its maximum height, (d) an expression for the position of
the body at any time t, and (e) the maximum height attained by the body.
7.58. Redo Problem 7.51 assuming there is air resistance which creates a force on the body equal to -2v Ib.

7.59. Redo Problem 7.54 assuming there is air resistance which creates a force on the body equal to ^-v Ib.

7.60. A ball of mass 5 slugs is dropped from a height of 1000 ft. Find the limiting velocity of the ball if it encounters a
force due to air resistance equal to —\v.

7.61. A body of mass 2 kg is dropped from a height of 200 m. Find the limiting velocity of the body if it encounters a
resistance force equal to —50v.

7.62. A body of mass 10 slugs is dropped from a height of 1000 ft with no initial velocity. The body encounters an air
resistance proportional to its velocity. If the limiting velocity is known to be 320 ft/sec, find (a) an expression for
the velocity of the body at any time t, (b) an expression for the position of the body at any time t, and (c) the time
required for the body to attain a velocity of 160 ft/sec.