How Does Physics Apply in the New Olympic Sled and Spring Event?

[Ajmathusek]

In a new Olympic event athletes run as fast as they can,
jump onto a sled, ride it down a hill and compress a
spring as far as they possibly can. (who thinks of these
any ways)
a. Consider a 55.0 kg athlete that makes it to a top
speed of 12.5 m/s before jumping onto a 15.0 kg sled. What is the athlete/sled initial speed as
she starts down the hill?
b. If the hill is 25.0 m long at an angle of 15.0 0 how much mechanical energy does the athlete/sled
initially have?
c. If the hill has a coefficient of friction of 0.125, what is the speed she reaches at the bottom of
the hill, just before hitting the spring?
d. Assuming the spring is located just at the bottom, and the coefficient of friction is the same as
on the hill, how far could this athlete compress the spring if it has a coefficient of 1250 N/m?

I am not sure how to go about this question. I know that you need acceleration for initial speed but can't figure out how to find it with the numbers given.

 

[mfb]

You don't need an acceleration. This is just a "collision" between athlete and sled. Only in (d), you can (but do not have to) calculate an acceleration.

 

[Ajmathusek]

So how do I go about doing it?

 

[mfb]

With the formulas you know for collision processes...

After the collision with the sled both move with the same velocity. Is this an elastic or inelastic collision?

 

[paranormal]

I would approach this problem by using the principles of Newton's Laws of Motion and conservation of energy.

a. To find the initial speed of the athlete and the sled, we can use the conservation of momentum equation, which states that the initial momentum of the system (athlete + sled) is equal to the final momentum of the system. In this case, we can write it as:
(mass of athlete x initial speed of athlete) + (mass of sled x initial speed of sled) = (total mass of system x final speed of system)
Therefore, we can calculate the initial speed of the athlete and sled as:
(55.0 kg x 12.5 m/s) + (15.0 kg x 0 m/s) = (70.0 kg x initial speed of system)
Initial speed of system = 8.93 m/s

b. To find the initial mechanical energy of the athlete and sled, we can use the equation for kinetic energy, which is given by:
KE = (1/2) x mass x (speed)^2
Therefore, the initial mechanical energy of the athlete and sled can be calculated as:
KE = (1/2) x (70.0 kg) x (8.93 m/s)^2 = 2,496.8 J

c. To find the final speed of the athlete and sled at the bottom of the hill, we can use the conservation of energy, which states that the initial energy of the system is equal to the final energy of the system. In this case, the initial energy is the mechanical energy calculated in part b, and the final energy is the sum of mechanical energy and potential energy due to gravity. We can write this as:
Initial energy = Final energy
2,496.8 J = (1/2) x (70.0 kg) x (final speed)^2 + (70.0 kg) x (9.8 m/s^2) x (25.0 m) x sin(15.0 0)
Solving for the final speed, we get:
Final speed = 9.34 m/s

d. To calculate how far the athlete can compress the spring, we can use the equation for potential energy stored in a spring, which is given by:
PE = (1/2) x k x (compression)^2
Where k is the spring constant and compression is the distance the spring