Assuming resistance proportional to the square of the

Problem 30P In Jules Verne’s original problem, the projectile launched from the surface of the earth is attracted by both the earth and the moon, so its distance r(t) from the center of the earth satisfies the initial value problem where Me and Mm denote the masses of the earth and the moon, respectively; R is the radius of the earth and S = 384,400 km is the distance between the centers of the earth and the moon. To reach the moon, the projectile must only just pass the point between the moon and earth where its net acceleration vanishes. Thereafter it is “under the control” of the moon, and falls from there to the lunar surface. Find the minimal launch velocity v0 that suffices for the projectile to make it “From the Earth to the Moon.”

Problem 1P The acceleration of a Maserati is proportional to the difference between 250 km/h and the velocity of this sports car. If this machine can accelerate from rest to 100 km/h in 10 s, how long will it take for the car to accelerate from rest to 200 km/h?

Problem 2P Suppose that a body moves through a resisting medium with resistance proportional to its velocity v, so that dv/dt = ?kv. (a) Show that its velocity and position at time t are given by v(t) = v0e?kt and ________________ (b) Conclude that the body travels only a finite distance, and find that distance.

Problem 3P Suppose that a motorboat is moving at 40 ft/s when its motor suddenly quits, and that 10 s later the boat has slowed to 20 ft/s. Assume, as in Problem, that the resistance it encounters while coasting is proportional to its velocity. How far will the boat coast in all? Problem Suppose that a body moves through a resisting medium with resistance proportional to its velocity v, so that dv/dt = ?kv. (a) Show that its velocity and position at time t are given by v(t) = v0e?kt and ________________ (b) Conclude that the body travels only a finite distance, and find that distance.

Problem 4P Consider a body that moves horizontally through a medium whose resistance is proportional to the square of the velocity v, so that dv/dt = ?kv2. Show that and that Note that, in contrast with the result of Problem, x(t) ? + ? as t ? + ?. Which offers less resistance when the body is moving fairly slowly-the medium in this problem or the one in Problem? Does your answer seem consistent with the observed behaviors of x(t) as t ? + ?? Problem Suppose that a body moves through a resisting medium with resistance proportional to its velocity v, so that dv/dt = ?kv. (a) Show that its velocity and position at time t are given by v(t) = v0e?kt and ________________ (b) Conclude that the body travels only a finite distance, and find that distance.

Problem 6P Assume that a body moving with velocity v encounters resistance of the form dv/dl = ?kv3/2.Show that and that Conclude that under a 3/2-power resistance a body coasts only a finite distance before coming to a stop.

Problem 7P Suppose that a car starts from rest, its engine providing an acceleration of 10 ft/s2, while air resistance provides 0.1 ft/s2 of deceleration for each foot per second of the car’s velocity, (a)Find the car’s maximum possible (limiting) velocity, (b) Find how long it takes the car to attain 90% of its limiting velocity, and how far it travels while doing so.

Problem 5P Assuming resistance proportional to the square of the velocity (as in Problem 4), how far does the motorboat of Problem 3 coast in the first minute after its motor quits? REFERENCE PROBLEM 4 Consider a body that moves horizontally through a medium whose resistance is proportional to the square of the velocity v, so that dv/dt = - kv2. Show that And that Note that, in contrast with the result of problem 2, x(t) as . Which offers less resistance when the body is moving fairly slowly -- the medium in this problem of the medium in problem 2? Does your answer seem consistent with the observed behaviours of x(t) as ? REFERENCE PROBLEM 3 Suppose that a motorboat is moving at 40 ft/s when its motor suddenly quits, and that 10 s later the boat has slowed to 20 ft/s. Assume as in Problem 2, that the resistance encounters while coasting is proportional to its velocity. How far will the boat coast in all? REFERENCE PROBLEM 2 Suppose that a body moves through a resisting medium with resistance proportional to its velocity v, so that dv/dt = - kv. (a.) Show that its velocity and position time t are given by And (b) Conclude that the body travels only a finite distance, and find that distance.

Problem 8P Rework both parts of Problem, with the sole difference that the deceleration due to air resistance now is (0.001)u2 ft/s2 when the car’s velocity is v feet per second.

Problem 9P A motorboat weighs 32,000 lb and its motor provides a thrust of 5000 lb. Assume that the water resistance is 100 pounds for each foot per second of the speed v of the boat. Then It the boat starts from rest, what is the maximum velocity that it can attain?

Problem 10P A woman bails out of an airplane at an altitude of 10,000 ft, falls freely for 20 s, then opens her parachute. How long will it take her to reach the ground? Assume linear air resistance pv ft/s2, taking ? = 0.15 without the parachute and ? = 1.5 with the parachute. (Suggestion: First determine her height above the ground and velocity when the parachute opens.)

Problem 11P According to a newspaper account, a paratrooper survived a training jump from 1200 ft when his parachute failed to open but provided some resistance by Happing unopened in the wind. Allegedly he hit the ground at 100 mi/h after falling for 8 s. Test the accuracy of this account. (Suggestion: Find ? in Eq. by assuming a terminal velocity of 100 mi/h. Then calculate the time required to fall 1200 ft.)

Problem 12P It is proposed to dispose of nuclear wastes-in drums with weight W = 640 lb and volume 8 ft3—by dropping them into the ocean (?0 = 0). The force equation for a drum falling through water is where the buoyant force B is equal tothe weight (at 62.5 lb/ft3) of the volume ofwater displaced by the drum (Archimedes? principle) and FR is the force of water resistance, found empirically to be 1 lb for each foot per secondof the velocity of a drum. If the drums are likely to burst upon an impact of more than 75 ft/s, what is the maximum depth to which they can be dropped in the ocean without likelihood of bursting?

Problem 14P Integrate the velocity function in Eq. (13) to obtain the upward-motion position function given in Eq. (14) with initial condition y(0) = y0.

Problem 15P Separate variables in Eq. (15) and substitute to obtain the downward-motion velocity function given in Eq. (16) with initial condition ?(0) = ?0

Problem 13P Separate variables in Eq. (12) and substitute to obtain the upward-motion velocity function given in Eq. (13) with initial condition ?(0) = ?0.

Problem 16P Integrate the velocity function in Eq. (16) to obtain the downward-motion position function given in Eq. (17) with initial condition y(0) = y0.

Problem 17P Consider the crossbow bolt of Example, shot straight upward from the ground (y = 0) at time t = 0 with initial velocity ?0 = 49 m/s. Take g = 9.8 m/s2 and ? = 0.0011 in Eq. (12) Then use Eqs. (13) and (14) to show that the bolt reaches its maximum height of about 108.47 m in about 4.61 s.

Problem 18P Continuing Problem, suppose that the bolt is now dropped (?0 = 0) from a height of y0 = 108.47 m. Then use Eqs. (16) and (17) to show that it hits the ground about 4.80s later with an impact speed of about 43.49 m/s. Problem Consider the crossbow bolt of Example, shot straight upward from the ground (y = 0) at time t = 0 with initial velocity ?0 = 49 m/s. Take g = 9.8 m/s2 and p = 0.0011 in Eq. (12) Then use Eqs. (13) and (14) to show that the bolt reaches its maximum height of about 108.47 m in about 4.61 s. Equation 12: Equation 13: Equation 14: Equation 16: Equation 17:

Problem 19P A motorboat starts from rest (initial velocity ?(0) = ?0 = 0). Its motor provides a constant acceleration of 4 ft/s2, but water resistance causes a deceleration of ?2/400 ft/s2. Find ? when t= 10 s, and also find the limiting velocity as t ? + ? (that is, the maximum possible speed of the boat).

Problem 20P An arrow is shot straight upward from the ground with an initial velocity of 160 ft/s. It experiences both the deceleration of gravity and deceleration ?2/800 due to air resistance. How high in the air does it go?

Problem 23P Suppose that the paratrooper of Problem falls freely for 30 s with ? = 0.00075 before opening his parachute. How long will it now take him to reach the ground? Problem Suppose that ? = 0.075 (in fps units, with g = 32 ft/s2) in Eq. (15) for a paratrooper falling with parachute open. If he jumps from an altitude of 10,000 ft and opens his parachute immediately, what will be his terminal speed? How long will it lake him to reach the ground?

Problem 21P If a ball is projected upward from the ground with initial velocity ?0 and resistance proportional to ?2, deduce from Eq. (14) that the maximum height it attains is

Problem 22P Suppose that ? = 0.075 (in fps units, with g = 32 ft/s2) in Eq. (15) for a paratrooper falling with parachute open. If he jumps from an altitude of 10,000 ft and opens his parachute immediately, what will be his terminal speed? How long will it lake him to reach the ground?

Problem 24P The mass of the sun is 329,320 times that of the earth and its radius is 109 times the radius of the earth, (a) To what radius (in meters) would the earth have to be compressed in order for it to become a black hole—the escape velocity from its surface equal to the velocity c = 3 x 108 m/s of light? (b) Repeat part (a) with the sun in place of the earth.

Problem 25P (a)Show that if a projectile is launched straight upward from the surface of the earth with initial velocity v0 less than escape velocity then the maximum distance from the center of the earth attained by the projectile where M and R are the mass and radiusof the earth, respectively. (b) With what initial velocity u0must such a projectile be launched to yield a maximum altitude of 100 kilometers above the surface of the earth? (c) Find the maximum distance from the center of the earth, expressed in terms of earth radii, attained by a projectile launched from the surface of the earth with 90% of escape velocity.

Problem 26P Suppose that you are stranded—your rocket engine has failed—on an asteroid of diameter 3 miles, with density equal to that of the earth with radius 3960 miles. If you have enough spring in your legs to jump 4 feet straight up on earth while wearing your space suit, can you blast off from this asteroid using leg power alone?

Problem 27P (a) Suppose a projectile is launched vertically from the surfacer = R of the earth with initial velocity where k2 = 2GM. Then solve the differential equation dr/dt= k/ (from Eq. (23) in this section) explicitly to deduce that r(t) ? ? as t ? ?. ________________ (b) If the projectile is launched vertically with initial velocity v0 > , deduce that Why does it again follow that r(t) ? ? as t ? ??

Problem 29P Suppose that a projectile is fired straight upward from the surface of the earth with initial velocity . Then its height y(t) above the surface satisfies the initial value problem Substitute dv/dt = v(dv/dy) and then integrate to obtain for the velocity v of the projectile at height y. What maximum altitude does it reach if its initial velocity is 1 km/s?

Problem 28P (a) Suppose that a body is dropped (v0 = 0) from a distance ro > R from the earth’s center, so its acceleration is dv/dr = ?GM/r2. Ignoring air resistance, show that it reaches the height r < ro at time (Suggestion: Substitute r = r0 cos2 ? to evaluate (b) If a body is dropped from a height of 1000 km above the earth’s surface and air resistance is neglected, how long does it take to fall and with what speed will it strike the earth’s surface?