A 748 N student stands in the middle of a frozen pond having a radius of 5.3 m. He is unable to get to the other side because of a lack of friction between his shoes and the ice. To overcome this difficulty, he throws his 2.1 kg physics textbook horizontally toward the north shore at a speed of 4.8 m/s. How long does it take him to reach the south shore?
To solve this problem, we can use the principles of conservation of momentum and Newton's second law of motion. Let's go step by step:
Step 1: Calculate the initial momentum of the student and the textbook.
Momentum (p) = mass (m) * velocity (v)
The momentum of the student is given by:
Momentum of the student = (mass of the student) * (velocity of the student)
= (748 N) * (0 m/s) (since the student is stationary)
= 0 kg·m/s
The momentum of the textbook is given by:
Momentum of the textbook = (mass of the textbook) * (velocity of the textbook)
= (2.1 kg) * (4.8 m/s)
= 10.08 kg·m/s
Step 2: Use the principle of conservation of momentum to find the final velocity of the student and the textbook.
According to the principle of conservation of momentum, the total momentum before the throw is equal to the total momentum after the throw.
Initial momentum of the system = Final momentum of the system
Since the student is initially at rest, the initial momentum of the system is 0 kg·m/s.
Final momentum of the system = (momentum of the student) + (momentum of the textbook)
So, final momentum of the system = 0 + 10.08 kg·m/s
= 10.08 kg·m/s
Step 3: Calculate the final velocity of the student.
Final velocity of the student = Final momentum of the system / (mass of the student)
The mass of the student is not given directly, but the weight of the student is given as 748 N. We can use the equation: Weight (W) = mass (m) * acceleration due to gravity (g)
So, mass of the student = Weight of the student / acceleration due to gravity
= 748 N / 9.8 m/s^2 (acceleration due to gravity is approximately 9.8 m/s^2)
= 76.33 kg
Final velocity of the student = 10.08 kg·m/s / 76.33 kg
= 0.132 m/s
Step 4: Calculate the time taken by the student to reach the south shore.
Since the student throws the textbook horizontally toward the north shore, the final velocity of the student gives us the speed at which the student moves toward the south shore.
The distance from the center of the pond to the south shore is equal to the radius of the pond.
Time = Distance / Speed
So, Time = 5.3 m / 0.132 m/s
= 40.15 s
Therefore, it takes the student approximately 40.15 seconds to reach the south shore.
To find out how long it takes for the student to reach the south shore, we can analyze the motion of the textbook.
First, let's consider the horizontal motion of the textbook. Since there is no friction between the shoes and the ice, the horizontal motion remains uniform. This means that there is no acceleration in the horizontal direction, and the horizontal velocity of the textbook remains constant throughout its motion.
Given:
- Mass of the textbook (m) = 2.1 kg
- Initial velocity of the textbook (u) = 4.8 m/s
- Radius of the pond (r) = 5.3 m
Now, we need to determine the distance traveled by the textbook to reach the south shore. This distance will be equal to the circumference of the pond which can be calculated using the formula:
Circumference = 2 * π * radius
C = 2 * 3.14 * 5.3 m
C ≈ 33.38 m
Since the student is throwing the textbook in a horizontal direction, the distance traveled by the textbook is equal to the circumference of the pond. Therefore, the distance (s) is equal to 33.38 m.
We can use the formula of the uniformly accelerated linear motion to find the time (t) it takes for the textbook to reach the south shore:
s = u * t
Rearranging the formula, we have:
t = s / u
Substituting the values, we get:
t = 33.38 m / 4.8 m/s
t ≈ 6.96 s
Therefore, it takes approximately 6.96 seconds for the textbook to reach the south shore.