Evaluate the surface integral s f · ds for the given vector field f and the oriented surface s. in other words, find the flux of f across s. for closed surfaces, use the positive (outward) orientation. f(x, y, z) = y i − x j + z2 k s is the helicoid (with upward orientation) with vector equation r(u, v) = u cos v i + u sin v j + v k, 0 ≤ u ≤ 2, 0 ≤ v ≤ 5π

Asap and show ! a 14 kg rock starting from rest free falls through a distance of 5.0 m with no air resistance. find the momentum change of the rock caused by its fall and the resulting change in the magnitude of earths velocity. earth mass is 6.0 * 10^24 kg. show your work assuming the rock earth system is closed.

Arunner is jogging at a steady 3.6 km/hr. when the runner is 2.9 km from the finish line, a bird begins flying from the runner to the finish line at 14.4 km/hr (4 times as fast as the runner). when the bird reaches the finish line, it turns around and flies back to the runner. even though the bird is a dodo, we will assume that it occupies only one point in space, i.e., a zero-length bird. how far does the bird travel? (b) after this first encounter, the bird then turns around and flies from the runner back to the finish line, turns around again and flies back to the runner. the bird repeats the back and forth trips until the runner reaches the finish line. how far does the bird travel from the beginning? (i.e. include the distance traveled to the first encounter)

During a medieval siege of a castle, the attacking army uses a trebuchet to hurl heavy stones at the castle walls. if the trebuchet launches the stones with a velocity of at an angle of 50.0°, (a) how long does it take the stone to hit the ground? (b) what is the maximum distance that the trebuchet can be from the castle wall to be in range? (c) how high will the stones go? show all your work.