Question

A sprinter starts from rest and accelerates at a steady rate for the first $50m$ of a $100m$ race, and then continues at a constant velocity for the second $50m$ of the race. If the sprinter runs the $100m$ in a time of $10s$, what is his instantaneous velocity when he crosses the finish line?

• A. $5m/s$
• B. $10m/s$
• C. $12m/s$
• D. $15m/s$
• E. $20m/s$

Answer 1

The correct option is D $15m/s$

We know the total distance the sprinter covers, and we know the total time. However, since the acceleration isnt uniform, we cant calculate the velocity quite so simply. Rather, we need two equations, one for the first 5 0 meters of the race, and another for the second 5 0 meters. In the first 50 meters, the sprinter accelerates from an initial velocity of $v_0=0$ to a final velocity of v in an amount of time, $t_1$. We can express this relationship using the kinematic equation that leaves out velocity, and then solve for t:
$x=x_0+\frac{1}{2}(r+v_0)t_1$
$50m=\frac{1}{2}v{t}_1$
$t_1-\frac{100m}{v}$
In the last 5 0 meters of the race, the sprinter runs with a constant velocity of v , covering a distance of x = 50 m in a time $t_2$, Solving for, $t_2$ we find:
$t_2=\frac{50m}{v}$
We know that the total time of the race, $t_1+t_2=10s$. With this in mind, we can add the two sprint times together and solve for v :
$10s=\frac{100m}{v}+\frac{50m}{v}=\frac{150m}{v}$
$v=\frac{150m}{10s}=15m/s$