If the initial velocity of the bus is 5 m/s, what is the velocity of the bus at t = 3 seconds given that the acceleration is ax(t) = αt with α = 1.15 m/s³?

• A. 10.35 m/s
• B. 8.45 m/s
• C. 12.45 m/s
• D. 14.25 m/s

Answer: (A)

Verified step by step guidance

1. Start by identifying the given information: the initial velocity \( v_0 = 5 \text{ m/s} \), the acceleration function \( a_x(t) = \alpha t \) with \( \alpha = 1.15 \text{ m/s}^3 \), and the time \( t = 3 \text{ seconds} \).
2. To find the velocity at a specific time, use the formula for velocity with variable acceleration: \( v(t) = v_0 + \int_0^t a_x(t') \, dt' \).
3. Substitute the given acceleration function into the integral: \( v(t) = 5 + \int_0^t 1.15t' \, dt' \).
4. Evaluate the integral: \( \int_0^t 1.15t' \, dt' = 1.15 \left[ \frac{t'^2}{2} \right]_0^t = 1.15 \times \frac{t^2}{2} \).
5. Substitute \( t = 3 \) seconds into the expression for velocity: \( v(3) = 5 + 1.15 \times \frac{3^2}{2} \). Calculate this expression to find the velocity at \( t = 3 \) seconds.