A child on a sled is sliding at a speed of 4m/s across level ice. if the total mass of the sled + child is 50kg and the sled has steel runners, how long will it take the sled to come to a halt? what force will be required to start it moving again? If this force continues to be applied at what rate would the sled accelerate?

You seem to be missing coefficients of friction.

To determine how long it will take for the sled to come to a halt, we need to understand the concept of acceleration and deceleration.

When the sled is sliding, it experiences a deceleration force due to factors like friction with the ice and air resistance. This force opposes the motion of the sled and eventually brings it to a stop.

The key equation we can use here is Newton's second law of motion:

F = m * a

Where F is the force, m is the mass, and a is the acceleration of the sled.

At the point when the sled comes to a halt, the acceleration of the sled is zero (a = 0). Hence, we can rewrite Newton's second law as:

F = m * 0
F = 0

Therefore, no force is required to stop the sled once it is already in motion.

Now, let's consider the force required to start the sled moving again. To initiate the motion, there needs to be sufficient force to overcome the static friction between the sled's steel runners and the ice. The force required to overcome static friction is given by:

F_static_friction = μ_s * N

Where μ_s is the coefficient of static friction and N is the normal force acting on the sled. In this case, the normal force is equal to the weight of the sled and child, which can be calculated as:

N = m * g

Where g is the acceleration due to gravity (approximately 9.8 m/s^2).

To determine the coefficient of static friction for steel runners on ice, we would need additional information such as the temperature and condition of the ice surface. Different surfaces and conditions have different coefficients of friction.

Finally, to calculate the rate at which the sled would accelerate when the force is continuously applied, we can use Newton's second law again. Assuming the applied force is constant, the sled's acceleration can be determined using:

a = F / m

Where F is the applied force and m is the mass.

To provide a specific answer, we would need to know the magnitude of the applied force.