Solved: Two particles of equal mass m are at the vertices of the base of an equilateral

Roughly where is your center of mass when youre standing?

Explain why a high jumpers center of mass need not clear the bar.

The center of mass of a solid sphere is clearly at its center. If the sphere is cut in half and the two halves are stacked as in Fig. 9.23, is the center of mass at the point where they touch? If not, roughly where is it? Explain. Figure 9.23 For Thought and Discussion 3

The momentum of a system of pool balls is the same before and after they are hit by the cue ball. Is it still the same after one of the balls strikes the edge of the table? Explain.

An hourglass is inverted and placed on a scale. Compare the scale readings (a) before sand begins to hit the bottom; (b) while sand is hitting the bottom; and (c) when all the sand is on the bottom.

Why are cars designed so that their front ends crumple during an accident?

Give three everyday examples of inelastic collisions.

Is it possible to have an inelastic collision in which all the kinetic energy of the colliding objects is lost? If so, give an example. If not, why not?

If you want to stop the neutrons in a reactor, why not use massive nuclei like lead?

Why dont we need to consider external forces acting on a system as its constituent particles undergo a collision?

How is it possible to have a collision between objects that dont ever touch? Give an example of such a collision.

A pitched baseball moves no faster than the pitchers hand. But a batted ball can move much faster than the bat. Whats the difference?

Two identical satellites are going in opposite directions in the same circular orbit when they collide head-on. Describe their subsequent motion if the collision is (a) elastic or (b) totally inelastic.

A 28-kg child sits at one end of a 3.5-m-long seesaw. Where should her 65-kg father sit so the center of mass will be at the center of the seesaw?

Two particles of equal mass m are at the vertices of the base of an equilateral triangle. The triangles center of mass is midway between the base and the third vertex. Whats the mass at the third vertex?

Rework Example 9.1 with the origin at the center of the barbell, showing that the physical location of the center of mass doesnt depend on your coordinate system.

Three equal masses lie at the corners of an equilateral triangle of side L. Find the center of mass.

How far from Earths center is the center of mass of the Earth Moon system? (Hint: Consult Appendix E.)

A popcorn kernel at rest in a hot pan bursts into two pieces, with masses 91 mg and 64 mg. The more massive piece moves horizontally at 47 cm/s. Describe the motion of the second piece.

A 60-kg skater, at rest on frictionless ice, tosses a 12-kg snowball with velocity v ! = 53.0ni + 14.0nj m/s, where the x- and y-axes are in the horizontal plane. Find the skaters subsequent velocity.

A plutonium-239 nucleus at rest decays into a uranium-235 nucleus by emitting an alpha particle 14 He2 with kinetic energy 5.15 MeV. Find the speed of the uranium nucleus.

A toboggan of mass 8.6 kg is moving horizontally at 23 km/h. As it passes under a tree, 15 kg of snow drop onto it. Find its subsequent speed.

A 150-g trick baseball is thrown at 60 km/h. It explodes in flight into two pieces, with a 38-g piece continuing straight ahead at 85 km/h. How much energy do the pieces gain in the explosion?

An object with kinetic energy K explodes into two pieces, each of which moves with twice the speed of the original object. Find the ratio of the internal kinetic energy to the center-of-mass energy after the explosion.

Two 140-kg satellites collide at an altitude where g = 8.7 m/s2 , and the collision imparts an impulse of 1.8*105 N# s to each. If the collision lasts 120 ms, compare the collisional impulse to that imparted by gravity. Your result should show why you can neglect the external force of gravity.

High-speed photos of a 220@mg flea jumping vertically show that the jump lasts 1.2 ms and involves an average vertical acceleration of 100g. What (a) average force and (b) impulse does the ground exert on the flea during its jump? (c) Whats the change in the fleas momentum during its jump?

Youre working in mission control for an interplanetary space probe. A trajectory correction calls for a rocket firing that imparts an impulse of 5.64 N #s. If the rockets average thrust is 135 mN, how long should the rocket fire?

In a railroad switchyard, a 56-ton freight car is sent at 7.0 mi/h toward a 31-ton car moving in the same direction at 2.6 mi/h. (a) Whats the speed of the cars after they couple? (b) What fraction of the initial kinetic energy was lost in the collision?

In a totally inelastic collision between two equal masses, with one initially at rest, show that half the initial kinetic energy is lost.

A neutron (mass 1.01 u) strikes a deuteron (mass 2.01 u), and they combine to form a tritium nucleus (mass 3.02 u). If the neutrons initial velocity was 23.5ni + 14.4nj Mm/s and if the tritium leaves the reaction with velocity 15.1ni + 22.6nj Mm/s, what was the deuterons velocity?

Two identical trucks have mass 5500 kg when empty, and the maximum permissible load for each is 8000 kg. The first truck, carrying 3800 kg, is at rest. The second truck plows into it at 65 km/h, and the pair moves away at 27 km/h. As an expert witness, youre asked to determine whether the second truck was overloaded. What do you report?

An alpha particle 14 He2 strikes a stationary gold nucleus 1197Au2 head-on. What fraction of the alphas kinetic energy is transferred to the gold? Assume a totally elastic collision.

Playing in the street, a child accidentally tosses a ball at 18 m/s toward the front of a car moving toward him at 14 m/s. Whats the balls speed after it rebounds elastically from the car?

A block of mass m undergoes a one-dimensional elastic collision with a block of mass M initially at rest. If both blocks have the same speed after colliding, how are their masses related?

A proton moving at 6.9 Mm/s collides elastically head-on with a second proton moving in the opposite direction at 11 Mm/s. Find their subsequent velocities.

A head-on, elastic collision between two particles with equal initial speed v leaves the more massive particle 1m12 at rest. Find (a) the ratio of the particle masses and (b) the final speed of the less massive particle.

Find the center of mass of a pentagon with five equal sides of length a, but with one triangle missing (Fig. 9.24). (Hint: See Example 9.3, and treat the pentagon as a group of triangles.) a Figure 9.24 Problem 37

Wildlife biologists fire 20-g rubber bullets to stop a rhinoceros charging at 0.81 m/s. The bullets strike the rhino and drop vertically to the ground. The biologists gun fires 15 bullets each second, at 73 m/s, and it takes 34 s to stop the rhino. (a) What impulse does each bullet deliver? (b) Whats the rhinos mass? Neglect forces between rhino and ground.

Consider a system of three equal-mass particles moving in a plane; their positions are given by ai ni + bi nj, where ai and bi are bio functions of time with the units of position. Particle 1 has a1 = 6t 2 + 5 and b1 = 0; particle 2 has a2 = 4t + 3 and b2 = 4t; particle 3 has a3 = 8t and b3 = t + 4. Find the centerof-mass position, velocity, and acceleration of the system as functions of time.

Youre with 19 other people on a boat at rest in frictionless water. The groups total mass is 1500 kg, and the boats mass is 12,000 kg. The entire party walks the 6.5-m distance from bow to stern. How far does the boat move?

A hemispherical bowl is at rest on a frictionless counter. A mouse drops onto the bowls rim from a cabinet directly overhead. The mouse climbs down inside the bowl to eat crumbs at the bottom. If the bowl moves along the counter a distance equal to one-tenth of its diameter, how does the mouses mass compare with the bowls mass?

Physicians perform needle biopsies to sample tissue from internal organs. A spring-loaded gun shoots a hollow needle into the tissue; extracting the needle brings out the tissue core. A particular device uses 8.3-mg needles that take 90 ms to stop in the tissue, which exerts a stopping force of 41 mN. (a) Find the impulse imparted by the tissue. (b) How far into the tissue does the needle penetrate?

Find the center of mass of the uniform, solid cone of height h, base radius R, and constant density r shown in Fig. 9.25. (Hint: Integrate over disk-shaped mass elements of thickness dy, as shown in the figure.) R h }dy Figure 9.25 Problem 43

A firecracker, initially at rest, explodes into two fragments. The first, of mass 14 g, moves in the +x-direction at 48 m/s. The second moves at 32 m/s. Find the second fragments mass and the direction of its motion.

An 11,000-kg freight car rests against a spring bumper at the end of a railroad track. The spring has constant k = 0.32 MN/m. The car is hit by a second car of 9400-kg mass moving at 8.5 m/s, and the two couple together. Find (a) the maximum compression of the spring and (b) the speed of the two cars when they rebound together from the spring.

On an icy road, a 1200-kg car moving at 50 km/h strikes a 4400-kg truck moving in the same direction at 35 km/h. The pair is soon hit from behind by a 1500-kg car speeding at 65 km/h, and all three vehicles stick together. Find the speed of the wreckage.

A car of mass M is initially at rest on a frictionless surface. A jet of water carrying mass at the rate dm/dt and moving horizontally at speed v0 strikes the rear window of the car, which is at 45 to the horizontal; the water bounces off at the same speed, relative to the window, with which it hit (see Fig. 9.26). Find expressions for (a) the cars initial acceleration and (b) the maximum speed it reaches. Note: (b) doesnt require any calculation. It might help to sketch a rough plot (no calculations!) of the cars speed versus time. Figure 9.26 Problem 47

A 1250-kg car is moving with velocity v ! 1 = 36.2ni + 12.7nj m/s. It skids on a frictionless icy patch and collides with a 448-kg hay wagon with velocity v ! 2 = 13.8ni + 10.2nj m/s. If the two stay together, whats their velocity?

Masses m and 3m approach at the same speed v and undergo a head-on elastic collision. Show that mass 3m stops, while mass m rebounds at speed 2v.

A 238U nucleus is moving in the x-direction at 5.0*105 m/s when it decays into an alpha particle 14 He2 and a 234Th nucleus. The alpha moves at 1.4*107 m/s at 22 above the x-axis. Find the recoil velocity of the thorium.

A cylindrical concrete silo is 4.0 m in diameter and 30 m high. It consists of a 6000-kg concrete base and 38,000-kg cylindrical concrete walls. Locate the center of mass of the silo (a) when its empty and (b) when its two-thirds full of silage whose density is 800 kg/m3 . Neglect the thickness of the walls and base.

A 42-g firecracker is at rest at the origin when it explodes into three pieces. The first, with mass 12 g, moves along the x-axis at 35 m/s. The second, with mass 21 g, moves along the y-axis at 29 m/s. Find the velocity of the third piece.

A 60-kg astronaut floating in space simultaneously tosses away a 14-kg oxygen tank and a 5.8-kg camera. The tank moves in the x-direction at 1.6 m/s, and the astronaut recoils at 0.85 m/s in a direction 200 counterclockwise from the x-axis. Find the cameras velocity.

Assuming equal-mass pieces in Exercise 24, find the angles of the two velocities relative to the direction of motion before the explosion.

A 62-kg sprinter stands on the left end of a 190-kg cart moving leftward at 7.1 m/s. She runs to the right end and continues horizontally off the cart. What should be her speed relative to the cart so that once shes off the cart, she has no horizontal velocity relative to the ground?

Youre a production engineer in a cookie factory, where mounds of dough drop vertically onto a conveyor belt at the rate of one 12-g mound every 2 seconds. Youre asked to design a mechanism that will keep the conveyor belt moving at a constant 50 cm/s. What average force must the mechanism exert on the belt?

Mass m, moving at speed 2v, approaches mass 4m, moving at speed v. The two collide elastically head-on. Find expressions for their subsequent speeds.

Verify explicitly that kinetic energy is conserved in the collision of the preceding problem.

While standing on frictionless ice, you (mass 65.0 kg) toss a 4.50-kg rock with initial speed 12.0 m/s. If the rock is 15.2 m from you when it lands, (a) at what angle did you toss it? (b) How fast are you moving?

Youre an accident investigator at a scene where a drunk driver in a 1600-kg car has plowed into a 1300-kg parked car with its brake set. You measure skid marks showing that the combined wreckage moved 25 m before stopping, and you determine a frictional coefficient of 0.77. What do you report for the drunk drivers speed just before the collision?

A fireworks rocket is launched vertically upward at 40 m/s. At the peak of its trajectory, it explodes into two equal-mass fragments. One reaches the ground 2.87 s after the explosion. When does the second reach the ground?

Two objects moving in opposite directions with the same speed v undergo a totally inelastic collision, and half the initial kinetic energy is lost. Find the ratio of their masses.

Explosive bolts separate a 950-kg communications satellite from its 640-kg booster rocket, imparting a 350@N# s impulse. At what relative speed do satellite and booster separate?

Youre working in quality control for a model rocket manufacturer, testing a class-D rocket whose specifications call for an impulse between 10 and 20 N# s. The rockets burn time is t = 2.8 s, and its thrust during that time is F1t2 = at1t - t2, where a = -4.6 N/s2 . Does the rocket meet its specs?

Youre investigating an accident in which a 1040-kg Toyota Yaris and an 2140-kg Buick Enclave collided at right angles in an intersection. The combined wreckage skidded 12.3 m before stopping. You measure the coefficient of friction between tires and road and find it to be 0.712. Show that at least one car must have exceeded the 55-km/h speed limit at the intersection.

A 400-mg popcorn kernel is skittering across a nonstick frying pan at 8.2 cm/s when it pops and breaks into two equal-mass pieces. If one piece ends up at rest, how much energy was released in the popping?

Two identical objects with the same initial speed collide and stick together. If the composite object moves with half the initial speed of either object, what was the angle between the initial velocities?

A proton (mass 1 u) moving at 6.90 Mm/s collides elastically head-on with a second particle moving in the opposite direction at 2.80 Mm/s. After the collision, the proton is moving opposite its initial direction at 8.62 Mm/s. Find the mass and final velocity of the second particle.

Two objects, one initially at rest, undergo a one-dimensional elastic collision. If half the kinetic energy of the initially moving object is transferred to the other object, what is the ratio of their masses?

Blocks B and C have masses 2m and m, respectively, and are at rest on a frictionless surface. Block A, also of mass m, is heading at speed v toward block B as shown in Fig. 9.27. Determine the final velocity of each block after all subsequent collisions are over. Assume all collisions are elastic. v u A B C Figure 9.27 Problem 70

Derive Equation 9.15b.

An object collides elastically with an equal-mass object initially at rest. If the collision isnt head-on, show that the final velocity vectors are perpendicular.

A proton (mass 1 u) collides elastically with a stationary deuteron (mass 2 u). If the proton is deflected 37 from its original direction, what fraction of its kinetic energy does it transfer to the deuteron?

Two identical billiard balls are initially at rest when theyre struck symmetrically by a third identical ball moving with velocity v ! 0 = v0ni (Fig. 9.28). Find the velocities of all three balls after this elastic collision. v0 u Figure 9.28 Problem 74

Find an expression for the impulse imparted by a force F1t2 = F0 sin1at2 during the time t = 0 to t = p/a. Here a is a constant with units of s -1 .

A 32-u oxygen molecule 1O22 moving in the +x-direction at 580 m/s collides with an oxygen atom (mass 16 u) moving at 870 m/s at 27 to the x-axis. The particles stick together to form an ozone molecule. Find the ozones velocity.

A 114-g Frisbee is lodged on a tree branch 7.65 m above the ground. To free it, you lob a 240-g dirt clod vertically upward. The dirt leaves your hand at a point 1.23 m above the ground, moving at 17.7 m/s. It sticks to the Frisbee. Find (a) the maximum height reached by the Frisbee-dirt combination and (b) the speed with which the combination hits the ground.

You set a small ball of mass m atop a large ball of mass M W m and drop the pair from height h. Assuming the balls are perfectly elastic, show that the smaller ball rebounds to height 9h.

A car moving at speed v undergoes a one-dimensional collision with an identical car initially at rest. The collision is neither elastic nor fully inelastic; 5/18 of the initial kinetic energy is lost. Find the velocities of the two cars after the collision.

A 200-g block is released from rest at a height of 25 cm on a frictionless 30 incline. It slides down the incline and then along a frictionless surface until it collides elastically with an 800-g block at rest 1.4 m from the bottom of the incline (Fig. 9.29). How much later do the two blocks collide again? 1.4 m 25 cm 30 Figure 9.29 Problem 80

A 14-kg projectile is launched at 380 m/s at a 55 angle to the horizontal. At the peak of its trajectory it collides with a second projectile moving horizontally, in the opposite direction, at 140 m/s. The two stick together and land 9.6 km horizontally downrange from the first projectiles launch point. Find the mass of the second projectile.

During a crash test, a car moving at 50 km/h collides with a rigid barrier and comes to a complete stop in 200 ms. The collision force as a function of time is given by F = at4 + bt3 + ct2 + dt, where a = -8.86 GN/s4 , b = 3.27 GN/s3 , c = -362 MN/s2 , and d = 12.5 MN/s. Find (a) the total impulse imparted by the collision, (b) the average collisional force, and (c) the cars mass.

Use numerical or graphical techniques to estimate the peak force of the collision in the preceding problem, and determine when it occurs.

A block of mass m1 undergoes a one-dimensional elastic collision with an initially stationary block of mass m2. Find an expression for the fraction of the initial kinetic energy transferred to the second block, and plot your result for mass ratios m1/m2 from 0 to 20.

Two objects of unequal mass, one initially at rest, undergo a onedimensional elastic collision. For a given mass ratio, show that the fraction of the initial energy transferred to the initially stationary object doesnt depend on which object it is.

In Figure 9.6, the uniform semicircular wire has radius R. How far above the center of the semicircle is its center of mass?

Find the center of mass of a uniform slice of pizza with radius R and angular width u.

In a ballistic pendulum demonstration gone bad, a 0.52-g pellet, fired horizontally with kinetic energy 3.25 J, passes straight through a 400-g Styrofoam pendulum block. If the pendulum rises a maximum height of 0.50 mm, how much kinetic energy did the pellet have after emerging from the Styrofoam?

An 80-kg astronaut has become detached from the safety line connecting her to the International Space Station. Shes 200 m from the station, at rest relative to it, and has 4 min of air remaining. To get herself back, she tosses a 10-kg tool kit away from the station at 8.0 m/s. Will she make it back in time?

Astronomers detect extrasolar planets by measuring the slight movement of stars around the center of mass of the starplanet system. Considering just the Sun and Jupiter, determine the radius of the circular orbit the Sun makes about the SunJupiter center of mass.

A thin rod extends from x = 0 to x = L. It carries a nonuniform mass per unit length m = Mx a /L1+a , where M is a constant with units of mass, and a is a non-negative dimensionless constant. Find expressions for (a) the rods mass and (b) the location of its center of mass. (c) Are your results what you expect when a = 0?

Model rocket motors are specified by giving the impulse they provide, in N # s, over the entire time the rocket is firing. The table below shows the results of rocket-motor tests with different motors used to launch rockets of different masses. Determine two data-based quantities that, when plotted against each other, should give a straight line and whose slope should allow you to determine g. Plot the data, establish a best-fit line, and determine g. Assume that the maximum height is much greater than the distance over which the rocket motor is firing, so you can neglect the latter. Youre also neglecting air resistancebut explain how that affects your experimentally determined value for g. Impulse, J (N # s) 4.5 7.8 4.5 7.8 11 Rocket mass (g) (including motor) 180 485 234 234 485 Maximum height achieved (m) 22 13 19 51 23

A block of mass M is moving at speed v0 on a frictionless surface that ends in a rigid wall, heading toward a stationary block of mass nM, where n 1 (Fig. 9.30). Collisions between the two blocks or the left-hand block and the wall are elastic and onedimensional. (a) Show that the blocks will undergo only one collision with each other if n 3. (b) Show that the blocks will undergo two collisions with each other if n = 4. (c) How many collisions will the blocks undergo if n = 10, and what will be their final speeds? M nM v0 u Figure 9.30 Problem 93

Youre interested in the intersection of physics and sports, and you recognize that many sporting events involve collisionsbat and baseball, foot and football, hockey stick and puck, basketball and floor. Using strobe photography, you embark on a study of such collisions. Figure 9.31 is your strobe photo of a ball bouncing off the floor. The ball is launched from a point near the top left of the photo and your camera then captures it undergoing three subsequent collisions with the floor. The collisions between ball and floor are a. totally elastic. b. totally inelastic. c. neither totally elastic nor totally inelastic.

Youre interested in the intersection of physics and sports, and you recognize that many sporting events involve collisionsbat and baseball, foot and football, hockey stick and puck, basketball and floor. Using strobe photography, you embark on a study of such collisions. Figure 9.31 is your strobe photo of a ball bouncing off the floor. The ball is launched from a point near the top left of the photo and your camera then captures it undergoing three subsequent collisions with the floor. The fraction of the balls mechanical energy thats lost in the second collision is a. about 10%. b. a little less than half. c. a little more than half. d. about 90%.

Youre interested in the intersection of physics and sports, and you recognize that many sporting events involve collisionsbat and baseball, foot and football, hockey stick and puck, basketball and floor. Using strobe photography, you embark on a study of such collisions. Figure 9.31 is your strobe photo of a ball bouncing off the floor. The ball is launched from a point near the top left of the photo and your camera then captures it undergoing three subsequent collisions with the floor. The component of the balls velocity whose magnitude is most affected by the collisions is a. horizontal. b. vertical. c. Both are affected equally.

Youre interested in the intersection of physics and sports, and you recognize that many sporting events involve collisionsbat and baseball, foot and football, hockey stick and puck, basketball and floor. Using strobe photography, you embark on a study of such collisions. Figure 9.31 is your strobe photo of a ball bouncing off the floor. The ball is launched from a point near the top left of the photo and your camera then captures it undergoing three subsequent collisions with the floor. Compared with the time between bounces, the duration of each collision is a. a tiny fraction of the time between bounces. b. a significant fraction of the time between bounces. c. much longer than the time between bounces.