Math Problem Statement
A motorcycle is following a car that is traveling at constant speed on a straight highway. Initially, the car and the motorcycle are both traveling at the same speed of 19.5 m/s , and the distance between them is 90.0 m . After t1 = 3.00 s , the motorcycle starts to accelerate at a rate of 4.00 m/s2 . The motorcycle catches up with the car at some time t2 How long does it take from the moment when the motorcycle starts to accelerate until it catches up with the car? In other words, find t2−t1 .
Solution
We need to find the time it takes for the motorcycle to catch up with the car after the motorcycle starts accelerating.
Given:
- Initial speed of both the car and the motorcycle:
- Initial distance between the car and the motorcycle:
- The motorcycle starts accelerating after .
- Acceleration of the motorcycle:
To find:
- The time , where is the time when the motorcycle catches up with the car after the motorcycle starts accelerating.
Step-by-step solution:
-
Position of the car: The car is moving at a constant speed, so its position at any time after can be described by:
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Position of the motorcycle: After the motorcycle starts accelerating at , its position at any time after it starts accelerating is: where is the acceleration.
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Catching up condition: The motorcycle catches up to the car when their positions are the same: Substituting the expressions for and into this equation gives:
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Solve for : Since cancels out on both sides, we're left with: Substituting the given values and , we have: Simplifying: Taking the square root:
Final answer:
The time it takes for the motorcycle to catch up with the car after it starts accelerating is approximately .
Would you like more details or have any questions?
Here are some related questions you could explore:
- What is the distance traveled by the motorcycle while catching up?
- How fast is the motorcycle traveling when it catches up?
- What if the motorcycle accelerated at a different rate?
- How much total time does it take for the motorcycle to catch up, including the first 3 seconds?
- What if the initial distance was different?
Tip: When analyzing motion problems, breaking them into phases (constant speed, acceleration, etc.) makes solving much easier!
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Math Problem Analysis
Mathematical Concepts
Kinematics
Quadratic Equations
Relative Motion
Formulas
Position of the car: x_car(t) = v_0 * t
Position of the motorcycle: x_motorcycle(t) = v_0 * t + 1/2 * a * (t - t_1)^2
Catching up condition: x_motorcycle(t2) = x_car(t2)
Theorems
Equations of motion under constant acceleration
Suitable Grade Level
Grades 10-12
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