Math Problem Statement
You are 9.0 m from the door of your bus, behind the bus, when it pulls away with an acceleration of 1.0 m/s/s. You instantly start running toward the still-open door at 5.5 m/s. What is the maximum time you can wait before starting to run and still catch the bus?
Solution
To solve this problem, we need to determine the maximum time you can wait before starting to run and still catch the bus.
Step 1: Define Variables
- Distance between you and the bus initially:
- Your running speed:
- Acceleration of the bus:
- Time you wait before starting to run:
Step 2: Set Up Equations
-
The bus starts from rest with acceleration, so its position after seconds is given by:
-
You start running after a delay of . The time you spend running is , and since you're moving at constant speed, your position is:
Initially, you're 9.0 meters behind the bus, so the relative position between you and the bus must satisfy: Substituting the expressions for and :
Step 3: Solve for
We will now solve this equation for the maximum that ensures you catch the bus.
First, expand and rearrange the equation: Now, express : To find the maximum , we will need to determine the time at which the equation holds true by solving for when you and the bus meet.
I'll calculate this now.The times when you and the bus could meet are and . The relevant time for catching the bus is , since represents a scenario where you wouldn't catch up yet.
Step 4: Maximum Time Before Running
Using the time , we can now calculate the maximum time you can wait before running, which is . The equation for was: Substitute into the equation.
I'll compute this now.It seems like I can’t do more advanced data analysis right now. Please try again later.
In the meantime, if you'd like help understanding the final steps of the problem or want to try manually calculating , feel free to ask! Here's a reminder of the last equation you would need:
Let me know if you'd like further assistance with it or if you have any questions.
5 Related Questions:
- What happens if your running speed increases?
- How would the answer change if the bus had a different acceleration?
- What if the distance to the bus was greater than 9.0 m?
- How would air resistance affect the time to catch the bus?
- What if you started running before the bus began accelerating?
Tip:
Always keep track of units during calculations to ensure accuracy, especially in problems involving multiple physical quantities like speed and acceleration.
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Math Problem Analysis
Mathematical Concepts
Kinematics
Quadratic Equations
Relative Motion
Formulas
Position of the bus: x_bus(t) = 0.5 * a * t^2
Position of the runner: x_you(t) = v * (t - t_wait)
Relative position equation: x_you(t) = x_bus(t) + d_0
Theorems
Kinematic equations of motion
Quadratic formula
Suitable Grade Level
Grades 10-12
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