Question
Derive the velocity function \(v(t)\) from the acceleration \(a(t)= -kt\) given \(v(0)=v_0\). Use this velocity function to determine the stopping time, and then set up the definite integral for the stopping distance, representing the distance traveled from \(t=0\) until the car stops.
Hint
Hint
Integrate \(a(t)\) to find \(v(t)\). Set \(v(t) = 0\) to find the stopping time \(t_s\). The stopping distance is given by \(\int_{0}^{t_s} v(t)\,dt\).
-
Answer
-
Mark Scheme
-
Explanation
- Formula Booklet
-
Answer
Complete Answer
Velocity function: \(v(t) = v_0 – \frac{k t^2}{2}\). Stopping time: \(t_s = \sqrt{\frac{2v_0}{k}}\). Stopping distance integral: \(\int_{0}^{\sqrt{2v_0/k}} \Big(v_0 – \frac{k t^2}{2}\Big)\,dt\).
-
Mark Scheme
-
Answer Explanation