Question

A bus starts from rest, moves with a uniform acceleration ‘a'. Simultaneously a passenger at a distance x from the bus starts running to catch the bus. The minimum velocity of the passenger to catch the bus is​

Answer 1

Given:

A bus starts from rest, moves with a uniform acceleration ‘a'. Simultaneously a passenger at a distance x from the bus starts running to catch the bus.

To find:

The minimum velocity of the passenger to catch the bus is​?

Solution:

From given, we have,

A bus starts from rest.

⇒ u = 0 m/s

The bus moves with a uniform acceleration ‘a'.

⇒ a = a m/s²

Bus:

We use the formula,

S = ut + 1/2 at²

substituting the given values in above equation, so we have,

S = 0 + 1/2 at²

∴ S = 1/2 at²

Passenger:

We use the formula,

v = u + at

For the  passenger, we use the equation,

x + S = vt

⇒ x + 1/2at² = vt

⇒ at² - 2vt + 2x = 0

The above equation represents a quadratic equation,

To get real roots, the above equation should satisfy the condition,

(-2v)² - 4a(2x) ≥ 0

4v² ≥ 8ax

v² ≥ 2ax

⇒ v ≥ √2ax

Therefore, the minimum velocity of the passenger is √2ax.