a ball of mass 2 kg moves so that its position x as function of time t is given by x=t³/3 if x is measured in metres and 't' in seconds,what is the kinetic energy acquired by the particle in first two seconds ..?
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Answers
Position of the body varies as x=t^3\3
Thus the acceleration of the body at any instant=d^2x\dt^2=2t
Force any on body at any instant=ma=2mt
Thus work done on body in moving the object by a distance dx=F.dx=2mtdx
=2mt(t^2dt)
Thus total work done in 2s=∫ 0^2 2mt^3dt
=16J
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Answer :-
➞ 16 joule
Explanation :-
We are given,
- Mass of a ball, m = 2 kg
- Position of the ball is given as a function of time : x = t³ / 3
We have to find the kinetic energy acquired by the body in first two seconds.
But position is given as a function of time, so we differentiate it with respect to t to get velocity as a function of time.
⇒ x = t³ / 3
⇒ dx / dt = d(t³ / 3) / dt
⇒ v = 1/3 × 3 × t²
⇒ v = t²
Let us find the velocity at t = 0
⇒ v = (0)² => v = 0
We get to know that the ball was at rest at time t = 0, Now let us find the velocity of the ball after 2 seconds.
⇒ v = (2)²
⇒ v = 4 m/s [ x in meters, t in seconds]
Now, Let us find the kinetic energy acquired by the ball in two seconds.
⇒ KE = 1/2 (mv²)
⇒ KE = 1/2 × 2 × (4)²
⇒ KE = 4²
⇒ KE = 16 joule
Hence, The kinetic energy acquired by the body in two seconds is 16 joule.