The Physics
Hypertextbook
Opus in profectus

# Projectiles

## Problems

### practice

1. In the 1994 action-adventure film Speed, an extortionist equipped a Los Angeles bus with a bomb that was set explode if the speed of the bus fell below 50 mph (22 m/s). The police discovered the bomb and routed the bus on to a segment of freeway that was still under construction — their intention being to keep it out of the notoriously heavy Southern California traffic. In a twist of the plot, however, they overlooked a 50 foot (15 m) gap in the construction. Since the bomb would explode killing everyone on board if they slowed down, they decide to jump the gap. The missing segment lies on an effectively level stretch of elevated freeway, 50 feet (15 m) above the ground, yet somehow they manage to jump the gap at 67 mph (30 m/s) and land safely on the other side.
1. If the bus in Speed obeyed the laws of physics how would this scene end?
2. Just out of curiosity, what would happen if the bus in the movie Speed was driven off of a 15 m high, horizontal ramp with nothing but the ground to land on?
1. Where would the bus land?
2. What velocity would the bus have when it struck the ground? State its…
1. magnitude and
2. direction
3. Let's modify this stunt so that a bus like the one in the movie could jump over a 50 foot (15 m) gap in the freeway.
1. How long does it take to jump the gap traveling at the speed shown in the movie?
2. What upward velocity would a bus need to stay in the air this long?
3. What direction should the bus head?
4. What type of device is needed to make the bus travel in this direction?
2. Projectile launcher demo.
1. Launch horizontally from a convenient height above the floor (to a reproducible location).
1. Measure the horizontal distance traveled.
2. Measure the vertical height fallen.
3. Calculate the time to fall.
4. Calculate the horizontal velocity.
2. Launch from an interesting angle above the horizontal.
1. Measure the angle of inclination.
2. Determine the horizontal component of the projectile's initial velocity.
3. Determine the vertical component of the projectile's initial velocity.
4. Determine the time to reach the highest point.
5. Determine the height of the highest point (above the launch height).
6. Determine the location where the projectile returns to its launch height.
3. Shoot the monkey demo
1. max range 45°
2. max height 90°
3. max flight time 90°
4. max path length 56.46°
5. max area under curve 60°

### conceptual

1. trajectories-circular.pdf
The drawings on the accompanying pdf show a mass on the end of a string as it is spun counterclockwise in a vertical circle. A pair of scissors is used to cut the string cleanly and instantly at four different positions. Sketch the subsequent trajectory of the mass until it lands on the ground.
2. trajectories-pendulum.pdf
The drawings on the accompanying pdf worksheet show a pendulum as it swings from left to right. A pair of scissors is used to cut the string cleanly and instantly at four different positions. Sketch the subsequent trajectory of the mass until it lands on the ground.
3. When would a bird in the air not be considered a projectile? Under what circumstances could that same bird now become a projectile?

### numerical

1. While driving on the interstate one day at 27.8 m/s (100 km/h, 60.0 mph) I accidentally dropped the Encyclopedia of Physics out the window 1.15 m above the ground. Determine the following…
1. the horizontal and vertical components of the book's velocity the instant I released it
2. the time the book was in the air
3. the horizontal distance the book traveled before hitting the ground
4. the horizontal and vertical components of the book's velocity the instant it hit the ground
2. A ball rolls off the top step of a long staircase as shown in the diagram below. The rise and run of each step are 16 cm and 32 cm respectively.

Determine…
1. the step on which the ball lands if it rolled off the top step at 5.00 m/s
2. the ball's speed when it left the top step (stated as a range) if it landed on the seventh step
3. At the 1998 Punkin Chunkin World Championship, a pneumatically-driven device called the "Aludium Q36 Pumpkin Modulator" was able to project an 3.6–4.5 kg (8–10 pound) pumpkin intact for a total distance of 1227.23 m (4026.32 feet). What was the muzzle velocity (magnitude and direction) of the record-setting pumpkin?
4. Watch this video clip from the 1996 movie Rumble in The Bronx starring Jackie Chan before solving this problem. (Warning: This video contains some bad language and an unflattering image of a man's buttocks. Additional Warning: This movie was filmed almost entirely in Vancouver. Not a single frame was shot in the Bronx.) Jackie Chan jumped over a narrow one way street in Vancouver known as Trounce Alley or Blood Alley. Since I have not been there to measure any dimensions, we will have to assume it's wide enough to accommodate two cars side by side— about 5 m. This scene was shot from the top level of the parking structure in the abandoned Woodward's department store complex. Woodward's went out of business in 1993 and the parking structure was demolished in 2006. The only thing of known vertical size is Jackie Chan who is 1.74 m tall, but he is never vertical at any time during this jump. A wooden ramp is visible on the parking structure and appears to be in line with the roof of the the adjacent apartment building. We will have to assume the apartment building has floors separated by an appropriate height — about 3 m. Given these assumptions, answer the following questions.
1. How long was Jackie Chan in the air?
2. How fast would Jackie Chan have to run off the parking structure to perform this stunt?
3. Movies often exaggerate reality for dramatic effect. Is your answer to the previous part of this question a reasonable speed for an able-bodied person with stunt training? Justify your answer with physics. (Hint: If you have the time, you may want to read the Internet Movie Database entry for this film. It contains reliable descriptions of how this stunt was performed. You still need to justify your answer with physics, however.)
5. Bob Beamon's spectacular long jump of 8.90 m at the Mexico City Olympics of 1968 would last as a world record for 22 years. For reasons beyond the scope of this book, the optimal launch angle for a long jumper is 35.26° What was Bob Beamon's take off speed if we assume he left the ground at the optimal launch angle?
6. A baseball is batted. It's a long fly ball. 5.0 seconds later the ball reaches the outfield 100 meters away and returns to the height from which it left the bat.
1. What maximum height did the baseball reach?
2. Determine the horizontal component of the ball's initial velocity.
3. Determine the vertical component of the ball's initial velocity.
4. What is the magnitude of the ball's initial velocity?
5. At what angle did the ball leave the bat?
6. An outfielder is 20 meters away from the spot where the ball will land at the moment it leaves the bat. If he runs at 4.0 m/s will he be able to make in time to catch the ball?
7. For fun, I decide to drive my car off a ramp inclined 36.87° above the horizontal with a speed of 25.0 m/s. (Assume the takeoff and landing heights are identical and that air resistance is negligble.)
1. What are the horizontal and vertical components of my car's initial velocity?
2. How long was my car in the air?
3. What horizontal distance did my car travel before hitting the ground?
8. A baseball is batted. It's a long fly ball. 5.0 seconds later the ball reaches the outfield 100 meters away and returns to the height from which it left the bat.
1. What maximum height did the baseball reach?
2. Determine the horizontal component of the ball's initial velocity.
3. Determine the vertical component of the ball's initial velocity.
4. What is the magnitude of the ball's initial velocity?
5. At what angle did the ball leave the bat?
6. An outfielder is 20 meters away from the spot where the ball will land at the moment it leaves the bat. If he runs at 4.0 m/s will he be able to make in time to catch the ball?

### calculus

1. A ball is thrown from the ground into the air. When the ball is at a height of y = 9.1 m above the ground the velocity is observed to be…

v = (+7.6  +6.1 m/s

1. To what maximum height will the ball rise?
2. What is the total horizontal distance traveled by the ball?
3. What is the velocity of the ball (magnitude and direction) the instant before it hits the ground?

### statistical

1. horizontal-projectile.txt
A group of physics students recorded the range of a projectile launched horizontally from several different heights.
1. Compute the time it took the projetile to fall from the given heights.
2. Construct a graph of range vs. time. Add a line of best fit to the graph and use it to determine the initial velocity of the projectile.

### investigative

1. How high should a domed stadium be built so that no batted ball would ever strike the ceiling? Solve using the distance of the farthest batted ball.