Math Problem Statement
A self-driving car traveling along a straight section of road starts from rest, accelerating at 2.00 m/s2 until it reaches a speed of 26.0m/s. Then the vehicle travels for 51.0s at constant speed until the brakes are applied, stopping the vehicle in a uniform manner in an additional 5.00 s. How long is the self-driving car in motion (in s) What is the average velocity of the self-driving car for the motion described? (Enter the magnitude in m/s.)
Solution
To solve this, let's break the motion into three phases:
- Acceleration phase: The car accelerates from rest.
- Constant speed phase: The car travels at a constant speed.
- Deceleration phase: The car decelerates uniformly until it stops.
Phase 1: Acceleration
- Initial velocity () = 0 m/s (starts from rest)
- Final velocity () = 26.0 m/s
- Acceleration () = 2.00 m/s²
- Time taken () = ?
We use the formula:
Phase 2: Constant Speed
- Speed during this phase () = 26.0 m/s
- Time taken () = 51.0 s
Phase 3: Deceleration
- Initial velocity () = 26.0 m/s
- Final velocity () = 0 m/s (comes to a stop)
- Time taken () = 5.00 s
- Deceleration () = ?
We use the formula: Solving for : (Negative sign indicates deceleration)
Total time in motion
The total time the car is in motion is the sum of the time in all three phases:
Average velocity
To find the average velocity, we need to find the total distance traveled and divide it by the total time.
Distance in phase 1 (acceleration):
Using the equation:
Distance in phase 2 (constant speed):
Distance in phase 3 (deceleration):
Using the equation:
Total distance:
Average velocity:
Final answers:
- Total time in motion:
- Average velocity:
Would you like more details or clarifications?
Here are some related questions for further exploration:
- How far does the car travel during the acceleration phase alone?
- What would the stopping distance be if the deceleration was doubled?
- How would the average velocity change if the constant speed phase lasted longer?
- What is the total deceleration experienced during the stopping phase?
- How would the total time change if the car accelerated at a faster rate?
Tip: When solving motion problems, always break down the problem into different phases and apply the right kinematic equations for each phase.
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Math Problem Analysis
Mathematical Concepts
Kinematics
Acceleration
Constant Velocity
Deceleration
Average Velocity
Formulas
v = u + at
d = ut + (1/2)at²
v_avg = total distance / total time
Theorems
Kinematic equations of motion
Suitable Grade Level
Grades 10-12
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