Explanation
Solution:
Hint:
· Find the magnitude of velocity by applying the Kinetic energy formula.
· Find the distance for two revolutions.
· Apply third equation of motion to find the acceleration.
Step 1: Find the velocity of the particle from the given kinetic energy.
\dfrac{1}{2}m{v^2} = 8 \times {10^{ - 4}}
Here, m is the mass of the particle and v is the velocity.
\dfrac{1}{2} \times 10 \times {10^{ - 3}} \times {v^2} = 8 \times {10^{ - 4}}
v = 0.4\;m/s
Step 2: Write the total distance covered.
s = 2(2\pi r) ( particle covered two revolutions)
Here, s is the total distance covered.
Step 3: Apply the third equation of motion.
{v^2} -{u^2} = 2as (Where v is final velocity, u is initial velocity (given u=0), a is acceleration and s is distance)
\Rightarrow {(0.4)^2} = 2 \times a \times (4\pi \times 6.4 \times {10^{ - 2}})
\therefore a = 0.1\;m/{s^2}
Hence, option (a) is the correct answer.
The velocities are given as,
\dfrac{{dx}}{{dt}} = 6
{v_x} = 6
\dfrac{{dy}}{{dt}} = 6
{v_y} = 8 - 10t
The velocity at projection is given as,
v\left( t \right) = {v_x}\hat i + {v_y}\hat j
v\left( t \right) = 6\hat i + \left( {8 - 10t} \right)\hat j
v\left( 0 \right) = 6\hat i + 8\hat j
\left| {v\left( 0 \right)} \right| = \sqrt {{6^2} + {8^2}}
\left| {v\left( 0 \right)} \right| = 10\;{\rm{m/s}}
The velocity of projection is 10\;{\rm{m/s}}.
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