A 64-kg boy and a 42-kg girl use an elastic rope while engaged in a tug-of-war on an icy frictionless surface. If the acceleration of the girl toward the boy is 3.5 m/s2, determine the magnitude of the acceleration of the boy toward the girl.
F=F
ma=ma
64 * a = 42 * 3.5
a= 147/64 = 2.29
To determine the magnitude of the acceleration of the boy toward the girl, we can use Newton's second law of motion, which states that the force acting on an object is equal to the product of its mass and acceleration.
Let's assume that the force exerted by the girl on the rope is Fg, and the force exerted by the boy on the rope is Fb.
Based on the given information, we can set up the following equations:
For the girl: Fg = mg * ag
For the boy: Fb = mb * ab
Where:
mg = mass of the girl (42 kg)
ag = acceleration of the girl (-3.5 m/s^2, as acceleration is in opposite direction)
mb = mass of the boy (64 kg)
ab = acceleration of the boy (unknown)
Since the girl and the boy are connected by the rope, the forces they exert on the rope are equal in magnitude but opposite in direction. Therefore, Fg = -Fb.
So, we can rewrite the equations as follows:
mg * ag = -mb * ab
Now, we can solve for the magnitude of the acceleration of the boy (ab).
Rearranging the equation, we get:
ab = -(mg * ag) / mb
Substituting the given values into the equation:
ab = -((42 kg) * (-3.5 m/s^2)) / (64 kg)
Simplifying the equation:
ab = 2.28125 m/s^2
Therefore, the magnitude of the acceleration of the boy toward the girl is approximately 2.28 m/s^2.
To determine the magnitude of the acceleration of the boy toward the girl, we can use Newton's second law of motion. This law states that the force applied to an object is directly proportional to its mass and acceleration, and can be expressed using the formula:
Force = mass x acceleration
In this case, we have two forces acting on the boy and the girl: the force due to the boy's mass and the force due to the girl's mass. Since they are connected by an elastic rope and accelerating towards each other, the forces will be equal in magnitude but opposite in direction.
Let's assume the magnitude of the acceleration of the boy toward the girl is denoted as "a." The equation for the boy's force can be written as:
Force boy = mass boy x a
Similarly, the equation for the girl's force can be written as:
Force girl = mass girl x 3.5 m/s^2
Since the forces are equal in magnitude, we can set up the following equation:
mass boy x a = mass girl x 3.5 m/s^2
Plugging in the given masses, we have:
64 kg x a = 42 kg x 3.5 m/s^2
Now we can solve for the unknown magnitude of the acceleration of the boy toward the girl (a).