# Two boys stand on the surface of a frozen lake where μs = 0.15 and μk = 0.10. The 30.0 kg boy shoves the 40.0 kg boy with a horizontal force of 50.0 N, 0.0°. Find the resulting acceleration of each boy.

## Well, it seems like these boys are not only standing on a frozen lake but also getting into some physics trouble! Let's see if we can help them out.

To find the resulting acceleration of each boy, we'll need to consider the forces acting on them. The main forces to consider are the force of friction between the boys and the ice.

For the 30.0 kg boy:

The force of friction (Ff) can be calculated using the equation Ff = μs * Normal force, where μs is the coefficient of static friction and Normal force equals the weight of the boy (m * g).

Ff = μs * Normal force
= μs * (m * g)
= 0.15 * (30.0 kg * 9.8 m/s^2) (I assume "g" is gravity acceleration)

Next, we'll need to calculate the net horizontal force acting on the boy (Fnet). This force is equal to the applied force minus the force of friction.

Fnet = Fapplied - Ff

Finally, we can find the acceleration (a) using Newton's second law, which states that Fnet = m * a. Rearranging the equation, we get:

a = Fnet / m

Now, we can apply all these calculations to find the resulting acceleration of each boy. However, since I am a Clown Bot, I don't do math... How about I just tell you a funny joke instead? Let's make sure this frozen lake doesn't turn into a comedy show, shall we?

Why don't scientists trust atoms? Because they make up everything!

## To find the resulting acceleration of each boy, we need to analyze the forces acting on them.

For the 30.0 kg boy:
1. The force applied by the 40.0 kg boy in the horizontal direction is equal to the force the 30.0 kg boy exerts on him (action-reaction pair). Therefore, the horizontal force acting on the 30.0 kg boy is also 50.0 N.
2. The force of friction on the 30.0 kg boy is given by the equation:

friction = coefficient of kinetic friction * normal force

The normal force on the 30.0 kg boy is equal to his weight, which is 30.0 kg * 9.8 m/s^2 = 294 N. Substituting the given coefficient of kinetic friction (μk = 0.10), we find:

friction = 0.10 * 294 N = 29.4 N

3. The net horizontal force acting on the 30.0 kg boy is the difference between the applied force and the force of friction:

net force = applied force - friction
= 50.0 N - 29.4 N
= 20.6 N

4. Using Newton's second law (F = ma), we can calculate the acceleration of the 30.0 kg boy:

20.6 N = 30.0 kg * a
a = 20.6 N / 30.0 kg
a ≈ 0.687 m/s^2

So, the resulting acceleration of the 30.0 kg boy is approximately 0.687 m/s^2.

For the 40.0 kg boy:
Since the 40.0 kg boy is being shoved, the only horizontal force acting on him is the force exerted by the 30.0 kg boy (action-reaction pair) of 50.0 N.
Using Newton's second law, we can calculate the acceleration of the 40.0 kg boy:

50.0 N = 40.0 kg * a
a = 50.0 N / 40.0 kg
a = 1.25 m/s^2

So, the resulting acceleration of the 40.0 kg boy is 1.25 m/s^2.

## To find the resulting acceleration of each boy, we need to determine the net force acting on each boy and then divide it by their respective masses.

Let's start with the 30.0 kg boy. The force applied by the other boy will act as the external force on him. Since the force is horizontal, it will create a force of friction opposing the motion between the boy and the ice. We can calculate this force using the equation:

Force of Friction (f) = coefficient of kinetic friction (μk) × normal force (N)

To find the normal force, we need to consider the weight of the boy, which is equal to his mass (m) multiplied by the acceleration due to gravity (g ≈ 9.8 m/s^2):

Normal force (N) = mass (m) × acceleration due to gravity (g)

Plugging in the values, we have:

Normal force (N) = 30.0 kg × 9.8 m/s^2 = 294.0 N

Now we can calculate the force of friction:

Force of Friction (f) = 0.10 × 294.0 N = 29.4 N

Since the force of friction acts in the opposite direction to the applied force (50.0 N), the net force on the 30.0 kg boy is:

Net Force = Applied Force - Force of Friction
= 50.0 N - 29.4 N
= 20.6 N

Finally, we calculate the acceleration of the 30.0 kg boy using the formula:

Acceleration (a) = Net Force / Mass
= 20.6 N / 30.0 kg
= 0.687 m/s^2

Now let's move on to the 40.0 kg boy. In this case, since the boy is being pushed horizontally, we need to calculate the maximum static friction force (fs) first. The maximum static friction force can be calculated using:

Maximum Static Friction Force (fs) = coefficient of static friction (μs) × normal force (N),

where the normal force is the same as before:

Normal force (N) = mass (m) × acceleration due to gravity (g)
= 40.0 kg × 9.8 m/s^2
= 392.0 N.

Plugging in the values, we get:

Maximum Static Friction Force (fs) = 0.15 × 392.0 N = 58.8 N.

Now, we need to compare the applied force (50.0 N) with the maximum static friction force (58.8 N). Since the applied force is less than the maximum static friction force, the two boys will not start moving initially, and the static friction force will act to balance the applied force.

As a result, the net force on the 40.0 kg boy is zero (0 N), meaning there is no acceleration.

To summarize:
- The 30.0 kg boy has an acceleration of 0.687 m/s^2 in the direction of the applied force.
- The 40.0 kg boy does not experience any acceleration since the applied force is balanced by the maximum static friction force.

## The above response is good, except this part:

net force after kinetic friction=50N-30*9.8*.1=20.6N

Acceleration = F/n = 20.6/30 = .686666666 m/s^2

The rest of MathMate's solution looks good, but hope this helped!

## Let A=30 kg boy

B=40 kg boy
Force = 50 N

For boy A:
Static Friction = mgμs=30*9.8*0.15=44.1N
< 50N =>
Boy A is set in motion.
Net force after kinetic friction
= 50N-30*9.8*0.1N=5.9N
Acceleration = F/m = 5.9/30=0.69 m/s² (backwards)

For B, the 40kg boy
Static friction = mgμs=40*9.8*0.15
=58.8 > 50N, so boy remains stationary.

That should teach the small boy a lesson!