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The speed of a particle of mass m varies with the distance x as v(x) = ax -n. Assume v(x = O) = 0 at t = 0. (a) Find the force F(x) responsible. (b) Determine x(t) and

The speed of a particle of mass m varies with the distance x as v(x) = ax -n. Assume v(x = O) = 0 at t = 0. (a) Find the force F(x) responsible. (b) Determine x(t) and

Grade:6

1 Answers

Kapil Khare
80 Points
5 years ago
(a) Velocity of the particle, v = ax-n
\implies                         (dv/dx) = a
Acceleration the particle = b = v(dv/dx)
\implies                                  b = (ax-n)(a)
\implies                                  b = a2x-an
So force, F(x) = m*b
\implies       F(x) = m(a2x-an)
 
(b) Velocity of the particle = v = (dx/dt) = ax-n
\implies                                    [dx/(ax-n)] = dt
Integrating both sides, we get,
\implies                       {log[mod(ax-n)]}/a = t + c 
Put x=0 at t=0
\implies                                      (log n)/a = c
\implies                       {log[mod(ax-n)]}/a = t + (log n)/a
So,                                                 x(t) = (neat + n)/a

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