Suppose a 64-kg boy and a 49-kg girl use a massless rope in a tug-of-war on an icy, resistance-free surface. If the acceleration of the girl toward the boy is 3.0 m/s2, find the magnitude of the acceleration of the boy toward the girl.
Well, it sounds like the boy and the girl are really making some smooth moves on that icy surface! Now, let's calculate the magnitude of the acceleration of the boy towards the girl.
We can use Newton's second law, which states that the force acting on an object is equal to its mass multiplied by its acceleration. In this case, the girl's mass is 49 kg, and her acceleration towards the boy is 3.0 m/s^2. So, the force acting on the girl is:
Force = mass * acceleration
Force = 49 kg * 3.0 m/s^2
Force = 147 kg*m/s^2
According to Newton's third law, the force the girl exerts on the boy is equal in magnitude but opposite in direction to the force the boy exerts on the girl. So, the force the boy exerts on the girl is also 147 kg*m/s^2.
Now, let's calculate the acceleration of the boy towards the girl. We already know his mass is 64 kg, and we can use Newton's second law again:
Force = mass * acceleration
147 kg*m/s^2 = 64 kg * acceleration
Solving for acceleration, we find:
acceleration = 147 kg*m/s^2 / 64 kg
acceleration ≈ 2.3 m/s^2
So, the magnitude of the acceleration of the boy towards the girl is approximately 2.3 m/s^2.
Now, let's hope they don't slip and slide right into each other!
To find the magnitude of the acceleration of the boy toward the girl, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Let's denote the mass of the boy as m1 (64 kg) and the mass of the girl as m2 (49 kg).
The net force acting on the boy is given by the force exerted on him by the girl, and similarly, the net force acting on the girl is given by the force exerted on her by the boy. Since the rope connecting them is massless, the magnitude of the force exerted on the boy by the girl is equal to the magnitude of the force exerted on the girl by the boy.
Let's denote the magnitude of the force exerted on the boy by the girl as F1, and the magnitude of the force exerted on the girl by the boy as F2.
According to Newton's second law, we can write:
F1 = m1 * a1
F2 = m2 * a2
where a1 is the acceleration of the boy and a2 is the acceleration of the girl.
Given that the acceleration of the girl toward the boy is 3.0 m/s^2, we have:
a2 = 3.0 m/s^2
Since the magnitude of the force exerted on the boy by the girl is equal to the magnitude of the force exerted on the girl by the boy, we have:
F1 = F2
Now, we can use the equation F = m * a to find the magnitude of the force exerted on each person:
F1 = m1 * a1
F2 = m2 * a2
Substituting the given values:
F1 = 64 kg * a1
F2 = 49 kg * 3.0 m/s^2
Since F1 = F2, we can equate the magnitudes:
64 kg * a1 = 49 kg * 3.0 m/s^2
Simplifying the equation:
64 kg * a1 = 147 kg m/s^2
Dividing both sides by 64 kg:
a1 = 147 kg m/s^2 / 64 kg
Simplifying further:
a1 = 2.2969 m/s^2
Therefore, the magnitude of the acceleration of the boy toward the girl is approximately 2.3 m/s^2.
To find the magnitude of the acceleration of the boy toward the girl, we can use Newton's second law of motion.
Newton's second law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, this can be expressed as:
F = m * a,
where F is the force, m is the mass, and a is the acceleration. In this case, we know the mass of the boy (64 kg) and the acceleration of the girl toward the boy (3.0 m/s²).
First, we need to find the force acting on the girl-boy system. Since the girl is tugging on the boy, the force acting on the girl is equal to the force acting on the boy but in the opposite direction. Therefore, we can write the equation as:
Force on girl = -Force on boy.
Now, we can substitute the mass and acceleration values into Newton's second law equation for both the girl and the boy:
Force on girl = m1 * a1,
Force on boy = m2 * a2.
Since the force on the girl is the negative of the force on the boy, we can write:
m1 * a1 = -m2 * a2.
Now we can substitute the known values into the equation. Let's label the girl as object 1 (m1 = 49 kg) and the boy as object 2 (m2 = 64 kg):
(49 kg) * (3.0 m/s²) = -(64 kg) * a2.
Now we can solve for a2, the acceleration of the boy toward the girl:
a2 = (49 kg * 3.0 m/s²) / 64 kg.
Evaluating the expression:
a2 = 147 kg * m/s² / 64 kg.
Performing the calculation:
a2 ≈ 2.297 m/s².
So, the magnitude of the acceleration of the boy toward the girl is approximately 2.297 m/s².