Chapter 8

\(\bullet\) A movie stuntman (mass 80.0 \(\mathrm{kg}\) ) stands on a window ledge 5.0 \(\mathrm{m}\) above the floor (Fig. \(8.45 ) .\) Grabbing a rope attached to a chandelier, he swings down to grapple with the movie's villain (mass 70.0 \(\mathrm{kg}\) ), who is standing directly under the chandelier. (Assume that the stuntman's center of mass moves downward 5.0 \(\mathrm{m} .\) He releases the rope just as he reaches the villain.) (a) With what speed do the entwined foes start to slide across the floor? (b) If the coefficient of kinetic friction of their bodies with the floor is \(\mu_{k}=0.250,\) how far do they slide?

\(\bullet\) Tennis, anyone? Tennis players sometimes leap into the air to return a volley. (a) If a 57 g tennis ball is traveling horizontally at 72 \(\mathrm{m} / \mathrm{s}\) (which does occur), and a 61 kg tennis player leaps vertically upward and hits the ball, causing it to travel at 45 \(\mathrm{m} / \mathrm{s}\) in the reverse direction, how fast will her center of mass be moving horizontally just after hitting the ball? (b) If, as is reasonable, her racket is in contact with the ball for \(30.0 \mathrm{ms},\) what force does her racket exert on the ball? What force does the ball exert on the racket?

Two identical 1.50 \(\mathrm{kg}\) masses are pressed against opposite ends of a light spring of force constant \(1.75 \mathrm{N} / \mathrm{cm},\) compress- ing the spring by 20.0 \(\mathrm{cm}\) from its normal length. Find the speed of each mass when it has moved free of the spring on a frictionless horizontal lab table.

A rifle bullet with mass 8.00 g strikes and embeds itself in a block with a mass of 0.992 kg that rests on a frictionless, horizontal surface and is attached to a coil spring. (See Figure \(8.47 . )\) The impact compresses the spring 15.0 \(\mathrm{cm} .\) Calibration of the spring shows that a force of 0.750 \(\mathrm{N}\) is required to compress the spring 0.250 \(\mathrm{cm}\) . (a) Find the magnitude of the block's velocity just after impact. (b) What was the initial speed of the bullet?

\(\bullet\) A 5.00 g bullet traveling horizontally at 450 \(\mathrm{m} / \mathrm{s}\) is shot through a 1.00 kg wood block suspended on a string 2.00 \(\mathrm{m}\) long. If the center of mass of the block rises a distance of 0.450 \(\mathrm{cm},\) find the speed of the bullet as it emerges from the block.

\(\bullet\) Jonathan and Jane are sitting in a sleigh that is at rest on frictionless ice. Jonathan's weight is \(800 \mathrm{N},\) Jane's weight is \(600 \mathrm{N},\) and that of the sleigh is 1000 \(\mathrm{N} .\) They see a poisonous spider on the floor of the sleigh and immediately jump off. Jonathan jumps to the left with a velocity (relative to the ice) of 5.00 \(\mathrm{m} / \mathrm{s}\) at \(30.0^{\circ}\) above the horizontal, and Jane jumps to the right at 7.00 \(\mathrm{m} / \mathrm{s}\) at \(36.9^{\circ}\) above the horizontal (relative to the ice). Calculate the sleigh's horizontal velocity (magnitude and direction) after they jump out.

Animal propulsion. Squids and octopuses propel them- selves by expelling water. They do this by taking the water into a cavity and then suddenly contracting the cavity, forcing the water to shoot out of an opening. A 6.50 \(\mathrm{kg}\) squid (including the water in its cavity) that is at rest suddenly sees a dangerous predator. (a) If this squid has 1.75 \(\mathrm{kg}\) of water in its cavity, at what speed must it expel the water to suddenly achieve a speed of 2.50 \(\mathrm{m} / \mathrm{s}\) to escape the predator'? Neglect any drag effects of the surrounding water. (b) How much kinetic energy does the squid create for this escape maneuver?

Forensic science. Forensic scientists can measure the muzzle velocity of a gun by firing a bullet horizontally into a large hanging block that absorbs the bullet and swings upward. (See Figure 8.49.) The meas ured maximum angle of swing can be used to calculate the speed of the bullet. In one such test, a rifle fired a 4.20 g bullet into a 2.50 kg block hanging by a thin wire 75.0 \(\mathrm{cm}\) long. causing the block to swing upward to a maximum angle of \(34.7^{\circ}\) from the vertical. What was the original speed of this bullet?

\(\bullet\) A 20.0 -kg lead sphere is hanging from a hook by a thin wire 3.50 m long, and is free to swing in a complete circle. Suddenly it is struck horizontally by a 5.00 -kg steel dart that embeds itself in the lead sphere. What must be the minimum initial speed of the dart so that the combination makes a com- plete circular loop after the collision?

\(\bullet\) A blue puck with mass 0.0400 \(\mathrm{kg}\) , sliding with a velocity of magnitude 0.200 \(\mathrm{m} / \mathrm{s}\) on a frictionless, horizontal air table, makes a perfectly elastic, head-on collision with a red puck with mass \(m,\) initially at rest. After the collision, the velocity of the blue puck is 0.050 \(\mathrm{m} / \mathrm{s}\) in the same direction as its initial velocity. Find (a) the velocity (magnitude and direction) of the red puck after the collision; and (b) the mass \(m\) of the red puck.

The structure of the atom. During \(1910-1911,\) Sir Ernest Rutherford performed a series of experiments to determine the structure of the atom. He aimed a beam of alpha particles (helium nuclei, of mass \(6.65 \times 10^{-27} \mathrm{kg}\) ) at an extremely thin sheet of gold foil. Most of the alphas went right through with little deflection, but a small percentage bounced directly back. These results told him that the atom must be mostly empty space with an extremely small nucleus. The alpha particles that bounced back must have made a head-on collision with this nucleus. A typical speed for the alpha particles before the collision was \(1.25 \times 10^{7} \mathrm{m} / \mathrm{s},\) and the gold atom has a mass of \(3.27 \times 10^{-25} \mathrm{kg}\) . Assuming (quite reasonably) elastic collisions, what would be the speed after the collision of a gold atom if an alpha particle makes a direct hit on the nucleus?

Rocket failure! Just as it has reached an upward speed of 5.0 \(\mathrm{m} / \mathrm{s}\) during a vertical launch, a rocket explodes into two pieces. Photographs of the explosion reveal that the lower piece, with a mass one-fourth that of the upper piece, was moving downward at 3.0 \(\mathrm{m} / \mathrm{s}\) the instant after the explosion. (a) Find the speed of the upper piece just after the explosion. (b) How high does the upper piece go above the point where the explosion occurred?

. A 7.0 \(\mathrm{kg}\) shell at rest explodes into two fragments, one with a mass of 2.0 \(\mathrm{kg}\) and the other with a mass of 5.0 \(\mathrm{kg}\) . If the heavier fragment gains 100 \(\mathrm{J}\) of kinetic energy from the explo- sion, how much kinetic energy does the lighter one gain?

. Accident analysis. A 1500 \(\mathrm{kg}\) sedan goes through a wide intersection traveling from north to south when it is hit by a 2200 kg SUV traveling from east to west. The two cars become enmeshed due to the impact and slide as one there after. On-the-scene measurements show that the coefficient of kinetic friction between the tires of these cars and the pave- ment is \(0.75,\) and the cars slide to a halt at a point 5.39 \(\mathrm{m}\) west and 6.43 \(\mathrm{m}\) south of the impact point. How fast was each car traveling just before the collision?

BIO Momentum and the squirting squid. An interesting use of "rocket power" is that used by cephalopods such as octopi and squids. These animals take in seawater and then squirt it out at high speed. \(\mathrm{A} 2.5\) -kg squid can expel 0.25 \(\mathrm{kg}\) of seawater \((\mathrm{in}\) a short burst of 0.20 \(\mathrm{s} )\) with a speed of 600 \(\mathrm{cm} / \mathrm{s}\) What is the momentum of one squirt of water? $$\begin{array}{l}{\text { A. } 1.2 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} \text { in the direction of the squirt }} \\ {\text { B. } 1.5 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} \text { in the direction of the squirt }} \\\ {\text { C. } 12 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} \text { in the direction of the squirt }} \\ {\text { D. } 15 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} \text { in the direction of the squirt }}\end{array} $$