The velocity and acceleration vectors of a particle undergoing circular motion are \( \vec{v} = 2i + 4j \, \text{m/s} \) and \( \vec{a} = 2i + 4j \, \text{m/s}^2 \) respectively at an instant of time. The radius of the circle is a
The velocity and acceleration vectors of a particle undergoing circular motion are \( \vec{v} = 2i + 4j \, \text{m/s} \) and \( \vec{a} = 2i + 4j \, \text{m/s}^2 \) respectively at an instant of time. The radius of the circle is a
Show Hint
- 1 m
- 2 m
- 3 m
- 4 m
The Correct Option is A
Solution and Explanation
Step 1: Use the relationship between velocity, acceleration, and radius in circular motion.
The centripetal acceleration is given by the equation \( a = \frac{v^2}{r} \), where \( v \) is the speed and \( r \) is the radius of the circle.
Step 2: Calculate the magnitude of velocity and acceleration.
Magnitude of \( \vec{v} = \sqrt{(2)^2 + (4)^2} = \sqrt{20} = 2\sqrt{5} \, \text{m/s} \)
Magnitude of \( \vec{a} = \sqrt{(2)^2 + (4)^2} = \sqrt{20} = 2\sqrt{5} \, \text{m/s}^2 \)
Step 3: Apply the formula for centripetal acceleration.
\( r = \frac{v^2}{a} = \frac{(2\sqrt{5})^2}{2\sqrt{5}} = 1 \, \text{m} \)
Final Answer: \[ \boxed{1 \, \text{m}} \]
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