Question

a sprinter is running a 100m long race. she begins at rest, then increases her speed steadily for 4.10s as she runs the first 14.0m of a race, then continues with a constant speed. what is her maximum speed? how long does she take to complete the entire race?

Explanation:

Step1: Find maximum speed in acceleration - phase

For the first part of the motion (constant - acceleration), we use the equation $x = v_0t+\frac{1}{2}at^{2}$. Since $v_0 = 0$, $x=\frac{1}{2}at^{2}$. We know $x = 14.0m$ and $t = 4.10s$. First, find the acceleration $a$.
$a=\frac{2x}{t^{2}}=\frac{2\times14.0}{4.10^{2}}=\frac{28}{16.81}\approx1.666m/s^{2}$. Then, use the equation $v = v_0+at$. Since $v_0 = 0$, $v = at$. So $v=1.666\times4.10\approx6.83m/s$. Another way is to use the fact that for uniformly - accelerated motion starting from rest, $x=\frac{0 + v}{2}t$ (average - speed formula). Solving for $v$, we get $v=\frac{2x}{t}=\frac{2\times14.0}{4.10}\approx6.83m/s$.

Step2: Find time for the second part of the motion

The distance of the second - part of the motion is $x_2=100 - 14.0 = 86.0m$. The speed in the second part is the maximum speed $v = 6.83m/s$. Using the equation $t=\frac{x}{v}$, the time for the second part $t_2=\frac{86.0}{6.83}\approx12.6s$.

Step3: Find total time for the race

The total time $t_{total}=t_1 + t_2$, where $t_1 = 4.10s$ and $t_2\approx12.6s$. So $t_{total}=4.10+12.6 = 16.7s$.

Answer:

Maximum speed: $6.83m/s$
Total time: $16.7s$