8.114 CALC A Variable-Mass Raindrop. In a rocket-propul- sion problem the mass is variable. Another such problem is a rain- drop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is dp dv dm Fext = + dt dt dt = Suppose the mass of the raindrop depends on the distance x that it has fallen. Then m kx, where k is a constant, and dm/dt = kv. This gives, since Fext = mg, dv mg = m + v(kv) dt Or, dividing by k, dv xgx + v² dt This is a differential equation that has a solution of the form v = at, where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero. (a) Using the proposed solution for v, find the acceleration a. (b) Find the distance the raindrop has fallen in t = 3.00 s. (c) Given that k = 2.00 g/m, find the mass of the raindrop at t = 3.00 s. (For many more intriguing aspects of this problem, see K. S. Krane, American Journal of Physics, Vol. 49 (1981), pp. 113-117.)

8.114 CALC A Variable-Mass Raindrop. In a rocket-propul- sion problem the mass is variable. Another such problem is a rain- drop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is dp dv dm Fext = + dt dt dt = Suppose the mass of the raindrop depends on the distance x that it has fallen. Then m kx, where k is a constant, and dm/dt = kv. This gives, since Fext = mg, dv mg = m + v(kv) dt Or, dividing by k, dv xgx + v² dt This is a differential equation that has a solution of the form v = at, where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero. (a) Using the proposed solution for v, find the acceleration a. (b) Find the distance the raindrop has fallen in t = 3.00 s. (c) Given that k = 2.00 g/m, find the mass of the raindrop at t = 3.00 s. (For many more intriguing aspects of this problem, see K. S. Krane, American Journal of Physics, Vol. 49 (1981), pp. 113-117.)

University Physics Volume 1

18th Edition

ISBN:9781938168277

Author:William Moebs, Samuel J. Ling, Jeff Sanny

Publisher:William Moebs, Samuel J. Ling, Jeff Sanny

Chapter9: Linear Momentum And Collisions

Section: Chapter Questions

Problem 9.10CYU: Check Your Understanding Notice that the mass of the air in the tank was neglected in the analysis...

See similar textbooks

Similar questions

Check Your Understanding Suppose you have a macroscopic salt crystal (that is, a crystal that is large enough to be visible with your unaided eye). It is made up of a huge number of unit cells. Is the center of mass of this crystal necessarily at the geometric center of the crystal?
A 2000-kg railway freight car coasts at 4.4 m/s underneath a grain terminal, which dumps grain directly down into the freight car. If the speed of the loaded freight car must not go below 3.0 m/s, what is the maximum mass of grain that it can accept?
(a) What is the mass of a large ship that has a momentum of 1.60109kgm/s, when the ship is moving at a speed of 48.0 km/h? (b) Compare the ship's momentum to the momentum of a 1100-kg artillery shell fired at a speed of 1200 m/s.
A model rocket engine has an average thrust of 5.26 N. It has an initial mass of 25.5 g, which includes fuel mass of 12.7 g. The duration of its burn is 1.90 s. (a) What is the average exhaust speed of the engine? (b) This engine is placed in a rocket body of mass 53.5 g. What is the final velocity of the rocket if it were to be fired from rest in outer space by an astronaut on a spacewalk? Assume the fuel burns at a constant rate.
Check Your Understanding What is the physical difference ( or relationship) between dmdt and dmgdt in this example?
On a snow covered road, a car with mass 1.1 x 10^3 kg collides head on with a van having a mass of 2.5 x 10^3 kg traveling at 8.0m/s. As a result of the collision the vehicles lock together and immediately come to rest. Calculate the speed pf the car just before the collision. Neglect friction
During the US Open, a 56 g tennis ball is hit at 12.0 m/s. Considering, determine the maximum number of meters the ball will reach within 1.2 s.
A 4.5 kg puffer fish expands to 40% of its mass by taking in water. When the puffer fish is threatened, it releases the water toward the threat to move quickly forward. What is the ratio of the speed of the puffer fish forward to the speed of the expelled water backwards?
a. Estimate the average acceleration magnitude, in ft/s^2, of the goose in the collision. b. Estimate the magnitude of the force, in lbf, exerted on the goose in the collision.
A 40 kg ball moving to the right at 5.0m/s collides head-on with the stationary 2.0 kg ball. If the collision is elastic, determine the direction and speed of each ball after the collision.
A roller skater with a mass of 'M' is moving along the ice with a speed of 6.0 m/s. She collides head-on with a stationary skater who has a mass of '2M.' After the collision, the two skaters grab onto each other (without spinning or losing their balance) and move with a speed of numerical answer. m/s. Enter a
b. The drawing shows a sulfur dioxide molecule, SO2. It consists of two oxygen atoms and a sulfuratom. A sulfur atom is twice as massive as an oxygen atom. Using this information and the dataprovided in the drawing, find the center of mass of the sulfur dioxide molecule. Express youranswers in nanometers (1 nm = 10_9 m).