After an accident, a 60.0 kg Inuit girl finds herself at rest on the frictionless ice of a frozen
lake, 15.0 m north of her stationary 150 kg sled. Fortunately she is connected to the sled
by a rope of negligible mass, so she begins hauling the rope in, maintaining a constant
tension of 50.0 N in it.
Determine:
4.1 the accelerations of the girl and the sled.
4.2 the total kinetic energy of the girl and sled system
(a) just before they meet; and
(b) just after they meet
Hello, I have correctly answered question (4.1) and have determined the accelerations of the girl and sled to be 0.83 m/s^2 South and 0.33 m/s^2 North respectively.
I am having trouble determining question (4.2). I think I should determine the velocities but am not sure
Accgirl=60/hermass
Accsled=150/sledmass
now to get time, 15=1/2 (accgirl+accsled)t^2 solve for t.
Knowing time, vfeach=acc*time
KE= 1/2 masseach*Vfinaleach^2
To determine the total kinetic energy of the girl and sled system, we need to first determine the velocities of the girl and sled just before and just after they meet.
Let's start with the velocities just before they meet. We can use the equations of motion to relate acceleration, initial velocity, time, and displacement.
For the girl:
Since she is at rest initially, her initial velocity (v_gi) is 0 m/s.
The final velocity (v_gf) can be found using the equation v_gf = v_gi + a_g * t, where a_g is the girl's acceleration and t is the time.
Given that the girl's acceleration is 0.83 m/s^2 South, we need to determine the time it takes for her to reach the sled.
For the sled:
Similarly, the sled is initially at rest, so its initial velocity (v_si) is 0 m/s.
The final velocity (v_sf) can be found using the equation v_sf = v_si + a_s * t, where a_s is the sled's acceleration and t is the time.
Given that the sled's acceleration is 0.33 m/s^2 North, we need to determine the time it takes for the girl to reach the sled.
To find the time it takes for the girl to reach the sled, we can use the equation of motion s = v_ti + (1/2) * a * t^2, where s is the displacement.
In this case, the displacement is the distance between the girl and the sled, which is given as 15.0 m.
Using this equation, we can solve for the time it takes for the girl to reach the sled:
15.0 m = 0 * t + (1/2) * 0.83 m/s^2 * t^2
Rearranging the equation, we get:
0.415 t^2 = 15.0
Solving for t, we find:
t = sqrt(15.0 / 0.415) ≈ 8.30 s
Now, we can calculate the velocities just before they meet:
For the girl:
v_gf = 0 + 0.83 m/s^2 * 8.30 s ≈ 6.88 m/s South
For the sled:
v_sf = 0 + 0.33 m/s^2 * 8.30 s ≈ 2.74 m/s North
Now that we have the velocities, we can calculate the kinetic energy.
(a) just before they meet:
The total kinetic energy (K) is the sum of the individual kinetic energies of the girl (K_g) and the sled (K_s).
For the girl:
K_g = (1/2) * m_g * v_gf^2, where m_g is the mass of the girl.
For the sled:
K_s = (1/2) * m_s * v_sf^2, where m_s is the mass of the sled.
Given the mass of the girl (m_g) as 60.0 kg and the mass of the sled (m_s) as 150 kg, we can calculate the individual kinetic energies.
K_g = (1/2) * 60.0 kg * (6.88 m/s)^2 ≈ 1432 J
K_s = (1/2) * 150 kg * (2.74 m/s)^2 ≈ 557 J
The total kinetic energy just before they meet is the sum of these individual kinetic energies:
K_total = K_g + K_s ≈ 1432 J + 557 J ≈ 1989 J
(b) just after they meet:
Once the girl and sled meet, they move as a combined system. The total kinetic energy just after they meet will be equal to the total kinetic energy just before they meet.
So, the total kinetic energy just after they meet will also be approximately 1989 J.
To determine the total kinetic energy of the girl and sled system, you are correct that you need to determine their velocities. Once you have the velocities, you can use the formula for kinetic energy to calculate it.
To find the velocities just before and just after they meet, you can use the following steps:
1. Calculate the distance the girl and sled travel while accelerating towards each other. The distance can be determined by the initial position of the girl (15.0 m north) and the position of the sled (0 m). Since they start at rest, both the girl and sled will travel the same distance.
2. Use the equation of motion for uniformly accelerated motion:
d = (1/2) * a * t^2
where d is the distance traveled, a is the acceleration, and t is the time. Rearrange the equation to solve for time:
t = sqrt((2 * d) / a)
Substitute the distance from step 1 and the acceleration for the girl to calculate the time it takes for them to meet.
3. Use the time calculated in step 2 to find the final velocity of both the girl and sled. Use the equation of motion:
v = a * t
Substitute the acceleration for each object and the time calculated from step 2 to determine the velocities just before and just after they meet.
4. Finally, calculate the total kinetic energy using the formula:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass of the system (girl + sled), and v is the velocity.
By following these steps, you should be able to determine the total kinetic energy of the girl and sled system just before and just after they meet.