The coefficient of kinetic friction for a 22 kg bobsled on a track is 0.20. What force is required to push it down a 6.0° incline and achieve a speed of 70 km/h at the end of 75 m?

To find the force required to push the bobsled down the incline and achieve the desired speed, we need to consider the following factors: the gravitational force, the frictional force opposing motion, and the net force required to accelerate the bobsled.

Step 1: Calculate the gravitational force acting on the bobsled
The gravitational force is given by the formula: F_gravity = m * g, where m is the mass of the bobsled and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Let's calculate F_gravity:
F_gravity = 22 kg * 9.8 m/s^2 = 215.6 N

Step 2: Determine the frictional force
The frictional force can be calculated using the formula: F_friction = μ * F_normal, where μ is the coefficient of kinetic friction and F_normal is the normal force acting on the bobsled.
The normal force can be calculated based on the angle of the incline: F_normal = m * g * cos(theta), where theta is the angle of the incline (6.0° in this case).
Let's calculate F_normal and F_friction:
F_normal = 22 kg * 9.8 m/s^2 * cos(6.0°) ≈ 209.43 N
F_friction = 0.20 * 209.43 N = 41.89 N

Step 3: Calculate the net force required
The net force is the sum of the gravitational force and the frictional force: F_net = F_gravity + F_friction.
Let's calculate F_net:
F_net = 215.6 N + 41.89 N = 257.49 N

Step 4: Determine the distance over which the force is applied
In this case, the distance is given as 75 m.

Step 5: Calculate the work done by the net force
The work done by the net force can be calculated using the formula: Work = Force * Distance * cos(theta), where theta is the angle of the incline.
Let's calculate the work done by the net force:
Work = 257.49 N * 75 m * cos(6.0°) ≈ 18818 J

Step 6: Calculate the final velocity of the bobsled
The work done by the net force is equal to the change in kinetic energy: Work = (1/2) * m * (v_final^2 - v_initial^2), where v_final is the final velocity and v_initial is the initial velocity (assumed to be zero).
Let's rearrange the equation to solve for v_final:
v_final^2 = (2 * Work / m) + v_initial^2
v_final^2 = (2 * 18818 J / 22 kg) + 0 ≈ 1709.82 m^2/s^2
v_final ≈ √(1709.82 m^2/s^2) ≈ 41.35 m/s

Step 7: Convert the final velocity to km/h
To convert the final velocity to km/h, we'll multiply it by a conversion factor: 1 m/s = 3.6 km/h.
Let's convert the final velocity:
v_final = 41.35 m/s * 3.6 km/h ≈ 148.86 km/h

Therefore, to achieve a final speed of approximately 70 km/h at the end of 75 m on a 6.0° incline with a coefficient of kinetic friction of 0.20, a force of approximately 257.49 N is required.