Acceleration-time graph of a particle shown, Work done by all the forces acting on the particle of mass ‘m’ in time interval t1, and t2, while a1 and a2 are the acceleration time t1 and t2.Then the value of work done is ma12nt12r24−t14 . Then the value of n is

v1=12a1t1,v2=12a2t2w=12mv22−v12w=12m1/2a2t22−1/2a1t12w=18ma22t22−a12t12from graph Tan⁡θ=a1t2=×a2t2 a1t2=a2t1a2=a1t2t1w=18ma12t22×t22−a12t12t12⇒w=18a12t24−a12t14t12⇒w=ma128t12t24−t14

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Acceleration-time graph of a particle shown, Work done by all the forces acting on the particle of mass ‘m’ in time interval t₁, and t₂, while a₁ and a₂ are the acceleration time t₁and t₂.Then the value of work done is $\frac{{ma}_{1}^{2}}{{nt}_{1}^{2}} [r_{2}^{4} - t_{1}^{4}]$ . Then the value of n is

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Detailed Solution

$v_{1} = \frac{1}{2} a_{1} t_{1}$ , $v_{2} = \frac{1}{2} a_{2} t_{2}$

$w = \frac{1}{2} m (v_{2}^{2} - v_{1}^{2})$

$w = \frac{1}{2} m [{(1 / 2 a_{2} t_{2})}^{2} - {(1 / 2 a_{1} t_{1})}^{2}]$

$w = \frac{1}{8} m [a_{2}^{2} t_{2}^{2} - a_{1}^{2} t_{1}^{2}]$

$from graph Tan θ = \frac{a_{1}}{t_{2}} = \times \frac{a_{2}}{t_{2}} a_{1} t_{2} = a_{2} t_{1}$

$a_{2} = \frac{a_{1} t_{2}}{t_{1}}$

$w = \frac{1}{8} m [\frac{a_{1}^{2} t_{2}^{2} \times t_{2}^{2} - a_{1}^{2} t_{1}^{2}}{t_{1}^{2}}] \Rightarrow w = \frac{1}{8} [\frac{a_{1}^{2} t_{2}^{4} - a_{1}^{2} t_{1}^{4}}{t_{1}^{2}}]$