A particle is kept at rest at the origin. A constant force `rarrF` starts acting on it at `t = 0 `. Find the speed of the particle at time t.

Explanation

The equation of motion is . `rarr dp / dt = rarr F`
As the particle starts from rest and the force is always in the same direction, the motion will be along this direction , Thus, we can write `dp/ dt = F `
or, `(int_(0)^(P) dp = int_(0)^(t) F dt)`
` p = Ft`
or, ` m_0 V / (sqrt 1-v^(2) / c^(2)) = Ft `
or, `m_(0) ^(2) V^(2) = F^(2) t^(2) - F^(2) t^(2) / C^(2) V^(2)`
V^(2) (m_(0)^(2) + F^(2) t^(2) / C^(2)) = F^(2) t^(2)`
or, `V = Ftc / (sqrt m_(0) ^(2) C^(2) + F^(2) t^(2)) `
Note from example (47.4) that however large t may be , V can never exceed c, No matter how long you apply a force , the speed of a particl will be less than the speed c.