Math Problem Statement
A car drives past a police radar.
Calculate the amount of acceleration of the car if the police radar measures = 5.1 m/s², an angular velocity = -0.22 rad/s when the car is at distance r = 56.34 m and the angle is θ = 42°.
Solution
We are given the following information:
- Acceleration measured by radar:
- Angular velocity:
- Distance (radius):
- Angle:
To calculate the acceleration of the car, we need to consider both the tangential and radial components of acceleration in polar coordinates.
Step 1: Components of acceleration
In polar coordinates, acceleration has two components:
-
Tangential acceleration : This is the component of acceleration in the direction of motion. It is given by: where is the angular acceleration.
-
Radial acceleration : This is the component of acceleration directed towards or away from the center of rotation. It is given by: where is the angular velocity.
Step 2: Calculate the radial acceleration
Given that and , we can calculate the radial acceleration:
Step 3: Calculate the tangential acceleration
Now, we know that the total acceleration of the car is a combination of both radial and tangential acceleration components. The total acceleration is measured as , so we can use the following relation: Substitute the known values:
Step 4: Conclusion
The tangential acceleration of the car is approximately , and the radial acceleration is approximately .
The total acceleration is the vector sum of these components, which matches the given .
Would you like more details or have any questions about the calculation?
Related Questions:
- How does the angular velocity affect the radial acceleration?
- What is the role of the angle in this calculation?
- How can the tangential and radial accelerations be used to calculate the total acceleration?
- What would happen if the angular velocity increased in this scenario?
- How can you derive angular acceleration from the given information?
Tip: Understanding the difference between radial and tangential acceleration is crucial when analyzing motion in polar coordinates.
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Math Problem Analysis
Mathematical Concepts
Polar Coordinates
Acceleration
Angular Velocity
Kinematics
Formulas
a_r = r * ω^2
a^2 = a_r^2 + a_t^2
Theorems
Radial and Tangential Components of Acceleration
Suitable Grade Level
Grades 10-12
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