Example #1: Projectile Motion with Energy

Example #2: Projectile Motion with Energy

Practice: You are practicing jumping as far as you can. In one attempt, you run and leave the floor with 7 m/s directed at an unknown angle. What maximum height do you reach if your speed at that point is 5 m/s? Ignore air resistance.

Practice: When you launch a 3-kg object from the ground with unknown initial speed directed at 37° above the x-axis, it hits the building shown below at 15 m above the ground with 25 m/s. Calculate the object’s launch speed.

Practice: A 3-kg box is nudged off the top of the path shown below, slides down, and is launched form the lower end of the path. The path is frictionless and its highest point is 10 m above the ground. The lower end is 2 m above the ground and makes 53° with the horizontal. Calculate the box’s speed:

(a) at the lowest point in the path;

(b) just before it leaves the path;

(c) at its highest point;

(d) just before it hits the ground.

An archer performs 20.0 J of work to stretch a bow, storing elastic potential energy in it (much like a spring). The archer then loads an arrow of mass 100 g into the stretched bow, and fires it at an angle of 30.0° above the horizontal. What is the horizontal component of the arrow’s velocity as it leaves the bow
A. 20.0 m/s
B. 0.63 m/s
C. 17.3 m/s
D. 10.0 m/s
E. 0.55 m/s

A 7 kg projectile is fired at an initial speed of 43 m/s, with a launch angle of 32°. What is the initial kinetic energy of the projectile? What is the kinetic energy of the projectile at its maximum height?

In the high jump, the kinetic energy of an athlete is transformed into gravitational potential energy without the aid of a pole.With what minimum speed must the athlete leave the ground in order to lift his center of mass 2.10 m and cross the bar with a speed of 0.65 m/s ?

A 0.40-kg ball is thrown with a speed of 8.5 m/s at an upward angle of 36 .What is its speed at its highest point?How high does it go? (Use conservation of energy.)

For its size, the common flea is one of the most accomplished jumpers in the animal world. A 2.10 mm -long, 0.510 mg critter can reach a height of 18.0 cm in a single leap.Neglecting air drag, what is the takeoff speed of such a flea?Calculate the kinetic energy of this flea at takeoff and its kinetic energy per kilogram of mass.If a 71.0 kg , 1.90 m -tall human could jump to the same height compared with his length as the flea jumps compared with its length, how high could he jump, and what takeoff speed would he need?In fact, most humans can jump no more than 60.0 cm from a crouched start. What is the kinetic energy per kilogram of mass at takeoff for such a 71.0 kg person?Where does the flea store the energy that allows it to make such a sudden leap?

A ball is thrown upward with an initial velocity of 15.0 m/s at an angle of 60.0 above the horizontal.Use energy conservation to find the balls greatest height above the ground.

A film of Jesse Owenss famous long jump in the 1936 Olympics shows that his center of mass rose 1.1 m from launch point to the top of the arc. What minimum speed did he need at launch if he was traveling at 6.5 m/s at the top of the arc?

A cannon tilted up at a 31.0 angle fires a cannon ball at 76.0 m/s from atop a
21.0 m -high fortress wall.What is the balls impact speed on the ground below?

In the figure, water balloons are tossed from the roof of a building, all with the same speed but with different launch angles.Which one has the highest speed when it hits the ground? Ignore air resistance.

A 2.8-kg block slides over the smooth, icy hill shown in the figure . The top of the hill is horizontal and 70 m higher than its base.What minimum
speed must the block have
at the base of the 70-m
hill to
pass over the pit at the far (righthand)
side of that hill?

A sled with rider having a combined mass of 125 kg travels over the perfectly smooth icy hill shown in the accompanying figure.How far does the sled land from the foot of the cliff?

A baseball is thrown from the roof of exttip{h}{h} = 22.1 m -tall building with an initial velocity of magnitude 12.1 m/s and directed at an angle of 53.1 above the horizontal.You may want to review (Page).For related problemsolving tips and strategies, you may want to view a Video Tutor Solution of Energy in projectile motion.
What is the speed of the ball just before it strikes the ground? Use energy methods and ignore air resistance.What is the answer for part (A) if the initial velocity is at an angle of 53.1
below the horizontal?If the effects of air resistance are included, will part (A) or (B) give the higher speed?

A 62-kg skier starts from rest at the top of a ski jump, point A in the figure , and travels down the ramp.If friction and air resistance can be neglected, determine her speed vB when she reaches the horizontal end of the ramp at B.Determine the distance s to where she strikes the ground at C.

It's been a great day of new, frictionless snow. Julie starts at the top of the 60° slope shown in the figure. At the bottom, a circular arc carries her through a 90 m turn, and she then launches off a 3.0-m {
m m}high ramp. How far horizontally is her touchdown point from the end of the ramp?

A projectile is fired at an upward angle of 48.0° from the top of a 130-m-high cliff with a speed of 180 m/s m m/s. What will be its speed when it strikes the ground below? (Use conservation of energy. Neglect air resistance.)

A 63 kg skier starts from rest at the top of a ski jump, point A in the figure, and travels down the ramp. If friction and air resistance can be neglected, determine(a) her speed vB when she reaches the horizontal end of the ramp at B(b) the distance s to
where she strikes the ground
at C

A projectile is fired at an initial speed of v 0 at a launch angle of θ. Using energy conservation, find the maximum height of the projectile.

A boy starts at rest and slides down a frictionless slide as shown in the figure. The bottom of the track is a height h above the ground. The boy then leaves the track horizontally, striking the ground at a distance d as shown. Using energy methods, determine the initial height H of the boy above the ground in terms of h and d.

A 20.0-kg cannonball is fired from a cannon with muzzle speed of 1 000 m/s at an angle of 37.08 with the horizontal. A second ball is fired at an angle of 90.08. Use the isolated system model to find(a) the maximum height reached by each ball and(b) the total mechanical energy of the ball–Earth system at the maximum height for each ball. Let y =0 at the cannon.

Review. A baseball outfielder throws a 0.150-kg baseball at a speed of 40.0 m/s and an initial angle of 30.0° to the horizontal. What is the kinetic energy of the baseball at the highest point of its trajectory?

A child of mass m starts from rest and slides without friction from a height h along a slide next to a pool (Fig. P8.27). She is launched from a height h/5 into the air over the pool. We wish to find the maximum height she reaches above the water in her projectile motion.(a) Is the child–Earth system isolated or nonisolated? Why?(b) Is there a nonconservative force acting within the system?(c) Define the configuration of the system when the child is at the water level as having zero gravitational potential energy. Express the total energy of the system when the child is at the top of the waterslide.(d) Express the total energy of the system when the child is at the launching point.(e) Express the total energy of the system when the child is at the highest point in her projectile motion.(f) From parts (c) and (d), determine her initial speed vi at the launch point in terms of g and h.(g) From parts (d), (e), and(f), determine her maximum airborne height ymax in terms of h and the launch angle u.(h) Would your answers be the same if the waterslide were not frictionless? Explain.