Kinematics with Constant Acceleration
A sprinter reaches a top speed of 12.5 m/s in the first 2.1 seconds of a race.
What was her acceleration?
How far did she travel in that time?
Estimate how much average horizontal force the ground applied to her (you will first have to estimate a mass).
The correct answer and explanation is :
Solution:
Finding the Acceleration
We use the kinematic equation:
[
v_f = v_i + at
]
where:
- ( v_f = 12.5 ) m/s (final velocity)
- ( v_i = 0 ) m/s (assuming she starts from rest)
- ( t = 2.1 ) s (time)
- ( a ) is the acceleration we need to find
Solving for ( a ):
[
a = \frac{v_f – v_i}{t} = \frac{12.5 – 0}{2.1} = \frac{12.5}{2.1} \approx 5.95 \text{ m/s}^2
]
Finding the Distance Traveled
Using the kinematic equation:
[
s = v_i t + \frac{1}{2} a t^2
]
Substituting the known values:
[
s = 0(2.1) + \frac{1}{2} (5.95)(2.1)^2
]
[
s = \frac{1}{2} (5.95) (4.41)
]
[
s \approx 13.1 \text{ m}
]
Finding the Average Horizontal Force
Using Newton’s Second Law:
[
F = ma
]
We estimate the sprinter’s mass as 60 kg:
[
F = (60)(5.95) = 357 \text{ N}
]
Explanation (300 words)
The sprinter starts from rest and reaches a top speed of 12.5 m/s in 2.1 seconds. To determine her acceleration, we use the kinematic equation ( v_f = v_i + at ). Given that her initial velocity is zero, solving for ( a ) gives an acceleration of approximately 5.95 m/s². This means her velocity increases by about 5.95 m/s every second.
To find how far she traveled during this acceleration phase, we use another kinematic equation, ( s = v_i t + \frac{1}{2} a t^2 ). Since she starts from rest, the equation simplifies to ( s = \frac{1}{2} a t^2 ). Substituting the values, we find that she covered 13.1 meters in these first 2.1 seconds.
Next, to estimate the average force the ground applied to her, we use Newton’s Second Law, ( F = ma ). Assuming the sprinter has a mass of about 60 kg (a reasonable estimate for an elite female sprinter), the force applied by the ground is approximately 357 N in the horizontal direction.
This force comes from the sprinter pushing backward against the ground, and in response, the ground applies an equal and opposite force forward, propelling her forward. The greater the force she applies, the greater the acceleration. Elite sprinters develop strong leg muscles to exert more force against the ground in a short time, allowing them to reach top speeds quickly.