Question

A passenger sees a bus standing 10 m ahead of him. The bus starts moving with a constant acceleration of \(5 \, m\,s^{-2}\) away from him. The passenger runs with constant speed to catch the bus. The minimum speed required to catch the bus is:

• A. \(10\sqrt{2}\, m\,s^{-1}\)
• B. \(5\sqrt{2}\, m\,s^{-1}\)
• C. \(10\, m\,s^{-1}\)
• D. \(50\, m\,s^{-1}\)

Hint:

For catch-up problems, always equate positions and apply the discriminant condition for minimum required speed.

Answer 1

The Correct Option is C

Step 1: Bus kinematics.
Starting 10 m ahead from rest with a = 5 m s⁻², bus displacement: x_b = 10 + ½·5·t² = 10 + 2.5t².

Step 2: Passenger kinematics.
Running at constant speed u, passenger displacement: x_p = u t.

Step 3: Catch condition.
Setting x_p = x_b gives u t = 10 + 2.5t² → 2.5t² - u t + 10 = 0.

Step 4: Real solution requirement.
The quadratic must have real roots, so discriminant u² - 4(2.5)(10) ≥ 0.

Step 5: Solving the inequality.
u² - 100 ≥ 0 → u² ≥ 100 → u ≥ 10 m s⁻¹.

Step 6: Conclusion.
The minimum speed required is 10 m s⁻¹.