How to solve this basic kinematics problem?

A sprinter runs the 100\ m floors in 10.0\ s. Approximate its motion assuming a constant acceleration in the first 15\ m and then a constant speed for the remaining 85\ m.

How to determinate:
1. its final speed
2. the time taken to cover the first 15\ m
3. the time needed for the other 85\ m
4. the magnitude of the acceleration for the first 15\ m

1 Answer
Andrea B.
May 21, 2018

Final speed
v=11.5 m/s
time 1: t_1 =2.6 s
time 2:t_2=7.4 s
acceleration:
a=4.41 m/s^2

Explanation:

You have 4 unknown data (t_1, t_2,v, a) and 4 equations:
t_1 + t_2 = 10.0 s
s_2=v xx t_2 =85 m for the second step
s_1=1/2 a xx t_1^2= 15 m for the first step
v=a xx t_1 for the first step

from the second equation, and by combining the third with the fourth equations, we obtain the time for the 2 paths
a=v/t_1
s_1=1/2 (v/t_1) xx t_1^2= 15 m
1/2 v xx t_1= 15 m
t_1 = 30/v
t_2= 85/v
that we put in the first equation
85/v + 30/v =10

now we have the speed
85+30 = 10 v
v=11.5 m/s
... the time 1
t_1 = 30/v= 30/11.5 =2.6 s
... the time 2
t_2 = 10-t_1 = 7.4 s
...the acceleration
a=v/t_1=(11.5 (m/s/))/(2.6s)=4.41 m/s^2

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