California Institute of Technology
Young Engineers & Science Scholars 2010
Problem Set 3: 2D and 3D Kinematics
Due: Wednesday, June 23, 2010
Problem 1
Two youngsters dive off of an overhang into a lake. Diver 1 drops straight down, whereas diver 2
runs off the cliff with an initial horizontal speed v0 .
(a) Find expressions for the splashdown speeds (i.e., the speed when hitting the water) for each
diver. Is the splashdown speed of diver 2 greater than, less than, or equal to the splashdown speed
of diver 1?
(b) What is the shape of the curved path followed by diver 2? Hint: Is it an arc of circle, a hyperbola, a parabola, or an arc of ellipse? Find y(x) and see what mathematical form it takes.
Problem 2
A trained dolphin leaps from the water with an initial speed of v0 . It jumps directly toward a ball
held by the trainer a horizontal distance d away and a vertical distance of h above the water. In the
absence of gravity the dolphin would move in a straight line to the ball and catch it, but because of
gravity the dolphin follows a path well below the ball’s initial position. If the trainer releases the
ball the instant the dolphin leaves the water, show that the dolphin and the falling ball meet above
water as long as h2 + d2 > 2hv02 /g.
Problem 3
A golfer wants to make a hole-in-one using his physics knowledge. The hole is located 50 m in front
of him, and 5 m to his right, as shown in the sketch. In addition to the acceleration due to gravity g,
there is a strong wind creating an acceleration w = 1.5 m/s2 in the positive z direction. The golfer
will hit the ball with an initial velocity in the x–y plane (i.e., directly in front of him), counting on
the wind to move the ball to his right so it falls in the right place.
(a) How long does the ball have to be in the air for the wind to be able to push it the necessary
distance of 5.0 m in the z direction?
(b) With what initial speed, v0 , and angle, θ, does the golfer have to hit the ball in order to make
a hole-in-one?
1
Problem 4: Challenge Problem (Optional)
A football kicker can give the ball an initial speed of 50 miles/hour. What are the (a) least and (b)
greatest elevation angles at which she can kick the ball to score a field goal from a point 50 yards
in front of goalposts whose horizontal bar is 10 feet above the ground? (c) Suppose the maximum
initial speed a kicker can give the ball is 50 miles/hour. Then what’s the longest field goal she can
hope to make? (d) At what elevation angle would she have to kick the ball in order to make her
longest-possible field goal?
2