Textbook Question
The position of an object is given by 𝓍 = At + Bt², where 𝓍 is in meters and t is in seconds. What is the acceleration as a function of time?
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2. 1D Motion / Kinematics
Intro to Acceleration
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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
Verified step by step guidance1
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 \( t = 3 \text{ seconds} \).
To find the velocity at a specific time, use the formula for velocity with variable acceleration: \( v(t) = v_0 + \int_0^t a_x(t') \, dt' \).
Substitute the given acceleration function into the integral: \( v(t) = 5 + \int_0^t 1.15t' \, dt' \).
Evaluate the integral: \( \int_0^t 1.15t' \, dt' = 1.15 \left[ \frac{t'^2}{2} \right]_0^t = 1.15 \times \frac{t^2}{2} \).
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.
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