A sprinter accelerates from rest to her top velocity at a rate of 3.80m/s2. After reaching her top velocity, she runs the remainder of the race at constant velocity. She completes the 50.00m long race in 7.88s . How far does the sprinter run while she is accelerating ?
Let's denote the acceleration time by Ta. Then
S = a*Ta²/2 + (a*Ta)*(T-Ta)
50 = 3.8*Ta²/2 + 3.8*Ta*(7.88-Ta);
1.9Ta² - 3.8Ta² + 29.944Ta - 50 = 0;
1.9Ta² - 29.944T + 50 = 0.
We've got a quadratic equation. It's solutions are
Ta1 ≈ 13.8615 Ta2 ≈ 1.8985
Ta1 is greater than 7.88, so we choose Ta = Ta2.
So, the sprinter runs Sa = a*Ta²/2 = 3.8*1.8985²/2 = 6.8480 m while she is accelerating.