A 25-g projectile is fired into a cube of ballistic gel at a velocity of 360 m/s. If the projectile penetrates 15 cm into the gel before stopping, find the average force exerted by the gel onto the projectile. Use

(b) the work-energy principle.

Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose one end of a rope with unstretched length ℓ is anchored to a cliff and a climber of mass m is attached to the other end. When the climber is a height ℓ above the anchor point, he slips and falls under the force of gravity for a distance 2ℓ , after which the rope becomes taut and stretches a distance x as it stops the climber (see Fig. 7–37). <IMAGE>

Assume a stretchy rope behaves as a spring with spring constant k.

(a) Applying the work-energy principle, show that

x = mg/k [ 1 + √1 + (4kℓ/mg)]

An airplane pilot fell 370 m after jumping from an aircraft without his parachute opening. He landed in a snowbank, creating a crater 1.1 m deep, but survived with only minor injuries. Assuming the pilot’s mass was 82 kg and his terminal velocity was 45 m/s, estimate:

(b) the average force exerted on him by the snow to stop him; and

Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose one end of a rope with unstretched length ℓ is anchored to a cliff and a climber of mass m is attached to the other end. When the climber is a height ℓ above the anchor point, he slips and falls under the force of gravity for a distance 2ℓ , after which the rope becomes taut and stretches a distance x as it stops the climber (see Fig. 7–37). <IMAGE>

Assume a stretchy rope behaves as a spring with spring constant k.

(b) Assuming m = 85kg , ℓ = 8.0 m and, k = 850 N/m determine x/ℓ (the fractional stretch of the rope) and kx / mg (the force that the rope exerts on the climber compared to his own weight) at the moment the climber’s fall has been stopped.

(II) How much work would be required to move a satellite of mass m from a circular orbit of radius r₁ = 2r_E about the Earth to another circular orbit of radius r₂ = 3r_E? (r_E is the radius of the Earth.)

(II) A car traveling at a velocity v can stop in a minimum distance d. What would be the car’s minimum stopping distance if it were traveling at a velocity of 2v?

(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.

(a) Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes.

(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)?