9. Work & Energy

(II) At an accident scene on a level road, investigators measure a car’s skid mark to be 78 m long. It was a rainy day and the coefficient of friction was estimated to be 0.30.

(b) Why does the car’s mass not matter?

(II) At an accident scene on a level road, investigators measure a car’s skid mark to be 78 m long. It was a rainy day and the coefficient of friction was estimated to be 0.30.

(c) What is wrong with a car that skids (see page 131)?

II) A 1.60-m-tall person lifts a 1.65-kg book off the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to

(c) How is the work done by the person related to the answers in parts (a) and (b)?

In the game of paintball, players use guns powered by pressurized gas to propel 33-g gel capsules filled with paint at the opposing team. Game rules dictate that a paintball cannot leave the barrel of a gun with a speed greater than 85 m/s. Model the shot by assuming the pressurized gas applies a constant force F to a 33-g capsule over the length of the 32-cm barrel. Determine F by

(a) using the work-energy principle, and

(II) A meteorite has a speed of 90.0 m/s when 750 km above the Earth. It is falling vertically (ignore air resistance) and strikes a bed of sand in which it is brought to rest in 3.25 m.

(b) How much work does the sand do to stop the meteorite (mass = 575 kg)?

(II) A NASA satellite is said to have observed an asteroid that is on a collision course with the Earth. The asteroid has an estimated mass, based on its size, of 4 x 10⁹ kg. It is approaching the Earth on a head-on course with a velocity of 600 m/s relative to the Earth and is now 5.0 x 10⁶ km away. With what speed will it hit the Earth’s surface, neglecting friction with the atmosphere?

We usually neglect the mass of a spring if it is small compared to the mass attached to it. But in some applications, the mass of the spring must be taken into account. Consider a spring of unstretched length ℓ and mass M_S uniformly distributed along the length of the spring. A mass m is attached to the end of the spring. One end of the spring is fixed and the mass m is allowed to vibrate horizontally without friction (Fig. 7–31). Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed v₀, the midpoint of the spring moves with speed v₀ / 2. <IMAGE>

Show that the kinetic energy of the mass plus spring when the mass m is moving with velocity v is

K = (1/2)Mv²

where M = m + (1/3)M_S is the “effective mass” of the system. [Hint: Let D be the total length of the stretched spring. Then the velocity of an infinitesimal length dx of spring, of mass dM, located at x is v(x) = v₀(x/D). Note also that dM = dx( M_S/D) .]

(II) A 3.5-kg object moving in two dimensions initially has a velocity v₁→ (10.0 î + 20.0 ĵ) m/s. A net force F→ then acts on the object for 2.0 s, after which the object’s velocity is v₂→ (15.0 î + 30.0 ĵ) m/s . Determine the work done by F→ on the object.

A 2.0-kg block slides across a rough surface with a constant coefficient of kinetic friction of 0.50 (Fig. 7–38a). The block starts at x= 0 with an initial velocity of 4.9 m/s. Pushing the block is a force directed at 36.8° below the horizontal and whose magnitude increases with position as shown in Fig. 7–38b.

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(d) Draw a line on the graph showing the magnitude of the friction force versus distance x.