There’s Been a Crime! Security Camera Footage: As experts in physics, you have been recruited by the local Police Department to help solve a recent crime Early this morning around 6:15 am, Wilma Wilshire, was preparing to transfer an empty antique vase that once belonged to the city’s first mayor, David Daisy. It was scheduled to be installed in the lobby of the Gardens Apartment Complex as part of a traveling antiques exhibit. Wilma parked around the corner of the apartment lobby’s front door, unwrapped the precious antique from its packaging, and started to walk it toward the entrance. Just before she made it through the lobby doors, she remembered that she left her van running. Since the street was pretty much deserted so early in the morning, she decided to set the vase down on the sidewalk for a moment while she ran to her van around the corner to turn off the engine. While she was walking back, she saw a blue car zip by and heard a loud crash. To her horror, she returned to find her priceless historic vase destroyed beyond repair and scattered among a mess of ceramic, potting soil, and red geraniums. The policeman that arrived first to the scene noticed a security camera outside of the ATM across the street and quickly requested the footage in hopes of catching the perpetrator in the act. Unfortunately, the camera footage seems worthless as the crash must have occurred in the exact moment that the blue car drove past. You can even see some of the debris flying through the air at this moment. The apartment manager noted that the red geraniums scattered about the crime scene appeared to be of the exact same variety as the potted plants that the apartment provided to each resident to place on their outdoor patio. Quick to remove himself from the scenario, he further explained that there is no way that a pot would have toppled on its own, it would have had to have been intentionally tampered with. Your task is to assemble clues and use evidence to figure out who is responsible for the destruction of this historic vase. Security Camera Footage:

Clue #1 The Scene of the Crime 1 cm = 2 m Wilma Wilshire’s Antiques Van Location of the Broken Vase Security Camera Traffic lights 1 cm = 2 m 2 4 6 8 10 m

Clue #2 The Car Trail… One of the investigators notices something and calls you over. It turns out that as the car captured in the security camera footage was accelerating from the traffic light, it was also dripping oil at a constant rate of 1 drop per second. A measurement scale has been overlaid to help you translate this diagram into quantifiable data. If the large puddle represents the location when the car was stopped at a stoplight, use this image of this scene to calculate the acceleration of this mystery car. 1 m

Clue #3 Apartment Balconies You ask the management for a blueprint and find that the lobby level is 4.5 m tall while each floor beyond that is 3 m tall. 16th Floor 3 m 15th Floor 14th Floor 13th Floor 12th Floor 11th Floor 10th Floor 9th Floor 8th Floor 7th Floor 6th Floor 5th Floor 4th Floor 3rd Floor 2nd Floor 1st Floor Lobby 4.5 m The railing of the apartment balcony is 1 meter above the floor level. Develop a math equation to calculate the balcony railing height for any level. Examples: The 5th floor balcony railing is 17.5 meters above ground level The 8th floor balcony railing is 26.5 meters above ground level

Clue #4 The Suspects After examining all of the balconies directly above the crime scene, you discover that the 8th, 12th, and 16th floor balconies are all missing their apartment provided potted plant, so you send someone to interview them. Here are their notes: Suspect #1 | Room 817 – Rachel Rose Very political. Tried to change the topic to the current mayor several times during the conversation Extremely messy apartment, had to clear a place to sit down for the interview Wilma Wilshire’s cousin Suspect #2 | Room 1217 – Don Daisy Had a hushed conversation on the phone while we were checking things out Has lived in this apartment for 14 years Well known in the community as a news anchor for the local news station Suspect #3 | Room 1617 – Freddy Flower Works late shift at the diner down the street and doesn’t get home until 3:00 am Recently moved in and doesn’t really have any connections in town yet. Claims to prefer staying indoors and doesn’t use the balcony so she gave her plant away

Clue #5 The Man in the Car After a little time, police were able to ID and track down the man in the car and bring him in for questioning. His name is Peter Petunia, and works as an Uber driver in the city so he is usually driving around early in the morning. At first it was believed that he might have seen something of interest since he drove by at the time of the crash, but after talking for a while, it was revealed that he was more involved than originally suspected. According to Peter, he had received a phone call from an unknown number the day before. This mystery caller had a series of detailed instructions for Peter to follow on his morning driving that day. The caller was very clear that after waiting at the stoplight outside of Gardens Apartments, Peter was supposed to accelerate constantly. He didn’t really know why but he seemed to remember hearing the caller say something about dropping something at the same time as the light turned green in order for the timing to work… You check out the car and in addition to the oil leak in the engine compartment, you notice that the middle of the passenger door is covered with dirt, ceramic pieces, and flower petals.

Clue #6 Drop Test 18 m/s 20 m/s 22 m/s 24 m/s Diagram of the sidewalk debris: You notice that on the sidewalk there is a significant amount of debris left from the impact of the plant. This debris pattern is diagramed on the right. You decide to do a little test to figure out if the debris pattern can tell you more information about the scenario. You simulate the impact by launching the same brand of potted plant towards the concrete at different impact velocities. Even though you aren’t able to perfectly recreate the pattern, you hope this will help identify the actual impact velocity. 0.9 m Data below taken from the lab testing at different impact velocities: 18 m/s 20 m/s 22 m/s 24 m/s 0.4 m 0.5 m 0.6 m 0.7 m

Evidence Sheet Evidence Sheet What does the evidence tell you? What do you know? How might this evidence help solve the mystery? What does the evidence tell you? What do you know? How might this evidence help solve the mystery?

Conclusion It is our professional opinion that the police should arrest Suspect #____ because

Solution – Car Timing 0 m/s --- 12 m 3 m/s2 2.83 s 12 meters From the oil drops, it can be determined that the car accelerates at about 3 m/s2 Time (s) Displacement (m) 0.0 1.0 1.5 2.0 6.0 3.0 13.5 𝑎= 2𝑑 𝑡 2 = 2 (13.5 𝑚) 3 𝑠 2 =3 𝑚 𝑠 2 It takes the car 2.83 seconds to travel the 12 meters to get in position 𝑣 𝑖 0 m/s 𝑣 𝑓 --- 𝑑 12 m 𝑎 3 m/s2 𝑡 2.83 s 𝑡= 2𝑑 𝑎 = 2 (12 𝑚) 3 𝑚 𝑠 2 =2.83 𝑠 *Note: this is the distance for the middle of the car to get there, not just the front 12 meters

Solution – Free Fall Timing Conversion for “Floor Number” to “Distance from Railing to Ground”: 16th Floor (50.5 m) 𝑑= 4.5 𝑚 + 𝑛−1 3 𝑚 +(1 𝑚) Lobby height Floor height Railing height 12th Floor (38.5 m) 𝑡= 2𝑑 𝑎 = 2 (38.5 𝑚) 10 𝑚 𝑠 2 =𝟐.𝟕𝟕 𝒔 8th Floor (26.5 m) 𝑣 𝑓 = 𝑣 𝑖 +𝑎𝑡= 0 𝑚 𝑠 + 10 𝑚 𝑠 2 2.77 𝑠 =𝟐𝟕.𝟕𝟓 𝒎 𝒔 Floor Height Hang Time Impact Velocity 16th Floor 50.5 m 3.18 s 31.78 m/s 12th Floor 38.5 m 2.77 s 27.75 m/s 8th Floor 26.5 m 2.30 s 23.02 m/s

Solution – Bringing it Together The evidence gained regarding the timing for the car and the impact velocity of the dropped object both suggest that the object was dropped from the 12th Floor making Suspect #2 – Don Daisy (the great great grandson of the vase’s original owner) our culprit Floor Height Hang Time Impact Velocity 16th Floor 50.5 m 3.18 s 31.78 m/s 12th Floor 38.5 m 2.77 s 27.75 m/s 8th Floor 26.5 m 2.30 s 23.02 m/s Car blocked the view roughly 2.8 seconds after the light turned green and the plant was dropped Impact velocity test in clue #6 suggests that the plant hit the ground going around 28 m/s 𝑣 𝑖 0 m/s 𝑣 𝑓 --- 𝑑 12 m 𝑎 3 m/s2 𝑡 2.83 s