Two skaters, a man and a woman, are standing on ice. Neglect any friction between the skate blades and the ice. The mass of the man is 85 kg, and the mass of the woman is 53 kg. The woman pushes on the man with a force of 58 N due east. Determine the acceleration (magnitude and direction) of (a) the man and (b) the woman.
To determine the acceleration of the man and the woman, we can use Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The equation for this law is:
F = ma
Where:
F is the net force acting on the object
m is the mass of the object
a is the acceleration of the object
(a) Let's start by finding the acceleration of the man. The net force acting on the man is the force exerted by the woman pushing him. This net force can be calculated by multiplying the force exerted by the woman (58 N) by the cosine of the angle between the force and the direction the man is being pushed (0 degrees since it is directly east).
F_man = F_woman * cos(0)
Since cos(0) = 1, we have:
F_man = 58 N
Using Newton's second law, we can rearrange the equation to solve for acceleration:
F_man = m_man * a_man
Substituting the known values, we can solve for a_man:
58 N = 85 kg * a_man
a_man = 58 N / 85 kg
a_man ≈ 0.6824 m/s²
So, the magnitude of the man's acceleration is approximately 0.6824 m/s² to the east.
(b) To find the acceleration of the woman, we can use the same approach. The net force acting on the woman is the force she exerts on the man, but in the opposite direction. So, its magnitude will be the same as the force exerted by the woman (58 N), but the opposite in direction.
F_woman = 58 N (opposite direction)
Similarly, using Newton's second law, we can solve for a_woman:
F_woman = m_woman * a_woman
Substituting the known values, we can solve for a_woman:
58 N = 53 kg * a_woman
a_woman = 58 N / 53 kg
a_woman ≈ 1.0943 m/s²
So, the magnitude of the woman's acceleration is approximately 1.0943 m/s² in the opposite (west) direction of the force she exerts on the man.