Where will they collide?

5. Intuitively, explain if the equation above supports the reasoning from your graph ( x v. t).

Within the space of a small Post-it note, determine where the two carts with meet. 

Two students, Ania and Britney, are observing two carts on a motion track. Cart 1's motion is described by the equation x = 0.12t², and Cart 2's motion is described by x = 1.5 - 0.30t², where x is the position in meters and t is the time in seconds. The graph above represents the motion of the two carts.

Ania: "Okay, so it looks like Cart 2 starts at the 1.5-meter mark, while Cart 1 starts at the origin. Since Cart 2's position is decreasing, it must be slowing down."

Britney: "That makes sense. And Cart 1 is definitely accelerating, since its graph is a curve. Because both carts are accelerating, they must have the same speed at the exact moment they are at the same position."

Your task is to act as a physics expert and analyze the conversation between Ania and Britney.

1. Analyze the Dialogue: For each student, identify which part of their statement is correct and which part is incorrect. Justify your answers using the principles of kinematics and evidence from the graph or equations.

2. Calculate the Collision: Ania and Britney are wondering at what exact time the carts collide. Set the two position equations equal to each other and solve for the time (t) when the carts are at the same position. Then, determine what that position (x) is.

Two students, Caitlin and Devon, are observing two carts on a motion track. Cart 1's motion is described by the equation x = 0.12t², and Cart 2's motion is described by x = 0.30t, where y is the position in meters and t is the time in seconds.

The Conversation:

Caitlin: "It's clear that both carts start from the same spot, the origin. Because Cart 1's graph is a curve, its speed must be greater than Cart 2's constant speed for the entire race."

Devon: "I agree. Cart 2 is definitely moving at a constant velocity since its graph is a straight line. When Cart 1 eventually catches up to Cart 2, their accelerations must be equal at that moment."

Your task is to act as a physics expert and analyze the conversation between Caitlin and Devon.

1. Analyze the Dialogue: For each student, identify which part of their statement is correct and which part is incorrect. Justify your answers using the principles of kinematics and evidence from the graph or equations.

2. Calculate the Collision: The carts are at the same position twice. One of those times is at the start (t=0). At what other time (t) do the carts have the same position? What is that position (x)?

Two students, Aashna and Oliva, are observing two carts on a motion track. Cart 1's motion is described by the equation x = 1.5 - 0.12t², and Cart 2's motion is described by x = 0.30t², where x is the position in meters and t is the time in seconds.

The Conversation:

Aashna: "Okay, so it looks like Cart 1 starts at the 1.5-meter mark and Cart 2 starts at the origin. Since Cart 1 has a negative acceleration, it must be slowing down for the entire time we are watching."

Oliva: "I agree that they are both accelerating since their graphs are curved. Because they start at different positions, the faster cart will have traveled a greater distance when they collide."

Student Task

Your task is to act as a physics expert and analyze the conversation between Aashna and Oliva.

1. Analyze the Dialogue: For each student, identify which part of their statement is correct and which part is incorrect. Justify your answers using the principles of kinematics and evidence from the graph or equations.

2. Calculate the Collision: At what time (t) do the carts have the same position? What is that position (x)?

Two students, Esmeralda and Scarlette, are observing two carts on a motion track. Cart 1's motion is described by the equation x = 1.75 - 0.12t², and Cart 2's motion is described by x = 0.30t, where x is the position in meters and t is the time in seconds.

The Conversation:

Esmeralda: "So, Cart 2 starts at the origin and moves away with a constant speed. Meanwhile, Cart 1 starts at the 1.75-meter mark and its speed decreases because it has a negative acceleration."

Scarlette: "That seems right. I noticed that Cart 1 starts from rest at its position. Since Cart 1 is accelerating and Cart 2 is not, Cart 1 must be moving faster than Cart 2 at the exact moment they collide."

Student Task

Your task is to act as a physics expert and analyze the conversation between Esmeralda and Scarlette.

1. Analyze the Dialogue: For each student, identify which part of their statement is correct and which part is incorrect. Justify your answers using the principles of kinematics and evidence from the graph or equations.

2. Calculate the Collision: At what time (t) do the carts have the same position? What is that position (x)? You will need to use the quadratic formula to solve this.