A 103 kg sprinter wishes to accelerate from rest to a speed of 13.2 m/s in a distance of 19.3 m.

What coefficient of static friction is required between the runners shoes and the track?

To find the coefficient of static friction required between the runner's shoes and the track, we can use the equation for acceleration:

v^2 = u^2 + 2as

where:
v = final velocity = 13.2 m/s
u = initial velocity = 0 m/s
a = acceleration
s = distance = 19.3 m

First, let's rearrange the equation to solve for acceleration:

a = (v^2 - u^2) / (2s)

Plugging in the given values:

a = (13.2^2 - 0^2) / (2 * 19.3)

a = 172.8 / 38.6

a = 4.474 m/s^2

Now, we can use the equation for the required static friction force:

F_friction = µ_s * m * g

where:
F_friction = required static friction force
µ_s = coefficient of static friction (what we're trying to find)
m = mass of the sprinter = 103 kg
g = acceleration due to gravity = 9.8 m/s^2

Rearranging the equation, we find:

µ_s = F_friction / (m * g)

To find F_friction, we can use Newton's second law of motion:

F_friction = m * a

Plugging in the given values:

F_friction = 103 kg * 4.474 m/s^2

F_friction = 461.222 N

Now, we can substitute this value back into the equation to find the coefficient of static friction:

µ_s = 461.222 N / (103 kg * 9.8 m/s^2)

µ_s ≈ 0.468

Therefore, the required coefficient of static friction between the runner's shoes and the track is approximately 0.468.

To determine the coefficient of static friction required between the sprinter's shoes and the track, we can use the equations of motion.

First, we need to calculate the sprinter's initial acceleration. We can use the first equation of motion:
vf^2 = vi^2 + 2ad, where
vf = final velocity = 13.2 m/s
vi = initial velocity = 0 m/s (since the sprinter starts from rest)
a = acceleration (unknown)
d = distance = 19.3 m

Substituting the values into the equation:
(13.2 m/s)^2 = (0 m/s)^2 + 2a(19.3 m)
173.16 m^2/s^2 = 38.6a

Now, solving for acceleration (a):
a = 173.16 m^2/s^2 / 38.6 m
a ≈ 4.48 m/s^2

Next, we can calculate the net force acting on the sprinter using Newton's second law of motion:
F_net = m * a, where
m = mass of the sprinter = 103 kg
a = acceleration = 4.48 m/s^2

Substituting the values:
F_net = 103 kg * 4.48 m/s^2
F_net ≈ 461.44 N

The net force acting on the sprinter is equal to the force of static friction between the sprinter's shoes and the track. Therefore, we can write:
F_friction = μ_s * m * g, where
F_friction = force of static friction
μ_s = coefficient of static friction (unknown)
m = mass of the sprinter = 103 kg
g = acceleration due to gravity = 9.8 m/s^2

Substituting the values:
461.44 N = μ_s * 103 kg * 9.8 m/s^2

Simplifying the equation:
461.44 N = 1009.4 μ_s

Now, solving for the coefficient of static friction (μ_s):
μ_s ≈ 461.44 N / 1009.4 kg
μ_s ≈ 0.457

Therefore, the coefficient of static friction required between the sprinter's shoes and the track is approximately 0.457.