Question

A particle starts with initial speed 'u' and retardation 'a' to come to rest in time T. Calculate the time taken to cover first half of the total path travelled.

Answer 1

Answer on Question #44163 – Engineering – Mechanics, Kinematics, Dynamics

A particle starts with initial speed 'u' and retardation 'a' to come to rest in time T. Calculate the time taken to cover first half of the total path travelled.

Solution:

u – initial speed of the particle;

a – retardation of the particle;

T – time of traveling;

t – time taken to cover first half of the total path;

Equation of motion for the particle (D – travelled distance):

$$D = u T - \frac{a T^2}{2}$$

Equation of motion for the particle for the first half of the path:

$$\frac{D}{2} = u t - \frac{a t^2}{2}$$

(1)in(2):

$$\frac{u T - \frac{a T^2}{2}}{2} = u t - \frac{a t^2}{2}$$ $$u T - \frac{a T^2}{2} = 2 u t - a t^2$$ $$a t^2 - 2 u t + T \left(u - \frac{a T}{2}\right) = 0$$

We have a quadratic equation and we need only positive root:

$$t = \frac{2 u + \sqrt{4 u^2 - 4 a T \left(u - \frac{a T}{2}\right)}}{2 a} = \frac{u + \sqrt{u^2 - a T \left(u - \frac{a T}{2}\right)}}{a}$$

Answer: time taken to cover first half of the total path: $t = \frac{u + \sqrt{u^2 - a T \left(u - \frac{a T}{2}\right)}}{a}$

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