A 38 kg girl and an 8.4 kg sled are on the frictionless ice of a frozen lake, 16 m apart bu connected by a rope of negligible mass. The girl exerts a horozontal 4.7 force on the rope.
(a) What is the acceleration of the girl?
m/s2
(b) What is the acceleration of the sled?
m/s2
(c) How far from the girl's initial position do they meet?
m
To solve this problem, we can apply Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass.
(a) Calculating the acceleration of the girl:
First, let's calculate the net force acting on the girl. Since there is no friction and the rope is assumed to be massless, the only force acting on the girl is the force she exerts on the rope.
So, the net force (F_net) is equal to the force exerted by the girl on the rope (4.7 N).
Next, we can apply Newton's second law to find the acceleration (a_girl) of the girl:
F_net = m_girl * a_girl
where m_girl is the mass of the girl (38 kg).
Rearranging the equation to solve for a_girl, we have:
a_girl = F_net / m_girl
Now, substituting the given values into the equation:
a_girl = 4.7 N / 38 kg
a_girl ≈ 0.1237 m/s²
Therefore, the acceleration of the girl is approximately 0.1237 m/s².
(b) Calculating the acceleration of the sled:
Since the girl and the sled are connected by the rope, they experience the same acceleration. Therefore, the acceleration of the sled (a_sled) is also 0.1237 m/s².
(c) Calculating the distance from the girl's initial position where they meet:
To find the distance they meet, we can use the equation of motion for the girl:
s_girl = u_girl * t + (1/2) * a_girl * t²
where s_girl is the distance traveled by the girl, u_girl is the initial velocity of the girl (assumed to be zero as she is at rest initially), a_girl is the acceleration of the girl, and t is the time taken.
Since the sled and the girl start 16 m apart, the total distance covered by both the girl and the sled is 16 m. Therefore, the distance traveled by the girl (s_girl) when they meet is equal to the initial separation distance minus the distance traveled by the sled (s_sled).
s_girl = 16 m - s_sled
From the equation of motion for the girl, we can rearrange it to solve for t:
t = √(2 * s_girl / a_girl)
Now, substituting the values into the equation, we get:
t = √(2 * (16 m - s_sled) / 0.1237 m/s²)
To solve for the distance (s_sled) where they meet, we need to know the value of time (t) or solve the equation further with another piece of information.
for the first two, F=ma will give the results.
for c), the cg of the system remains unchanged, so the initial cg is where they meet.