A 750 N student stands in the middle of a frozen pond having a radius of 5.4 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 2.0 kg physics textbook horizontally toward the north shore at a speed of 5.1 m/s. How long does it take him to reach the south shore?
To find the time it takes for the student to reach the south shore, we can use the principle of conservation of momentum.
1. Calculate the momentum of the student before throwing the textbook.
Momentum = mass × velocity
Momentum = 750 N × 0 m/s (since the student is stationary)
Momentum = 0 kg.m/s
2. Calculate the momentum of the textbook before it is thrown.
Momentum = mass × velocity
Momentum = 2.0 kg × 5.1 m/s
Momentum = 10.2 kg.m/s
3. Since momentum is conserved, the total momentum before and after the throw should be equal.
Momentum before throw = Momentum after throw
0 kg.m/s + 10.2 kg.m/s = (2.0 kg + mass of student) × velocity of student
4. Rearrange the equation to solve for the mass of the student.
10.2 kg.m/s = (2.0 kg + mass of student) × velocity of student
mass of student + 2.0 kg = 10.2 kg.m/s / 5.1 m/s
mass of student = (10.2 kg.m/s / 5.1 m/s) - 2.0 kg
mass of student = 2.0 kg
5. Now we can calculate the acceleration of the student using the formula:
Force = mass × acceleration
750 N = (2.0 kg + 2.0 kg) × acceleration
acceleration = 750 N / (4.0 kg)
acceleration = 187.5 m/s^2
6. Use the equation of motion (horizontal motion without friction) to find the time it takes for the student to reach the south shore. The equation is:
distance = initial velocity × time + (1/2) × acceleration × time^2
Since the initial velocity is zero (as the student is stationary), the equation simplifies to:
distance = (1/2) × acceleration × time^2
Rearrange the equation to solve for time:
time = sqrt(2 × distance / acceleration)
The distance to be covered is the radius of the pond, which is 5.4 m:
time = sqrt(2 × 5.4 m / 187.5 m/s^2)
time ≈ sqrt(0.0576 s^2 / m) ≈ 0.24 s
Therefore, it takes the student approximately 0.24 seconds to reach the south shore.
To find out how long it takes for the student to reach the south shore, we need to calculate the time it takes for the physics textbook to travel to the north shore, and then use this time to calculate the time for the student to reach the south shore.
First, let's find the initial horizontal velocity (Vx) of the textbook. Since the student throws it horizontally towards the north shore, there is no vertical component to its initial velocity, and thus its horizontal velocity is the same as the student's throwing velocity, which is 5.1 m/s.
Next, we can calculate the time it takes for the textbook to reach the north shore, using the formula:
Time (t) = Distance (d) / Velocity (v)
The distance that the textbook needs to travel is half the circumference of the frozen pond. Given that the pond has a radius of 5.4 m, the distance is:
Distance = 0.5 * (2 * π * radius)
Substituting the value of the radius into the equation:
Distance = 0.5 * (2 * 3.1415 * 5.4) = 17.0 m
Now, we can substitute the distance and velocity into the formula to find the time it takes:
t = 17.0 m / 5.1 m/s = 3.3 s
So it takes the textbook 3.3 seconds to reach the north shore.
Since the student throws the textbook, and they are in the middle of the pond, it will take the same amount of time for the student to reach the south shore. Therefore, the answer is 3.3 seconds.