How do I do these?
A passenger at an airport steps onto a moving sidewalk that is 100.0 m long and is moving at a speed of 1.5 m/s. The passenger then starts walking at a speed of 1.0 m/s in the same direction as the sidewalk is moving. What is the passenger's velocity relative to the following observers?
a. A person standing stationary alongside to the moving sidewalk
b. A person standing stationary on the moving sidewalk
c. A person walking alongside the sidewalk with a speed of 2.0 m/s and in a direction opposite the motion of the sidewalk
d. A person riding in a cart alongside the sidewalk with a speed of 5.0 m/s and in the same direction in which the sidewalk is moving.
e. A person riding in a cart with a speed of 4.0 m/s and in a direction perpendicular to the direction in which the sidewalk is moving.
Use the information given in item 3 to answer the following questions:
a. How long does it take for the passenger walking on the sidewalk to get from one end of the sidewalk to the other end?
b. How much time does the passenger save by taking the moving sidewalk instead of walking alongside it?
I will be happy to check or critique your thinking.
a. To find the passenger's velocity relative to a person standing stationary alongside the moving sidewalk, we can add the velocities of the passenger and the sidewalk. Since they are moving in the same direction, we can simply add the magnitudes (taking into account the direction).
The passenger's velocity = 1.0 m/s (direction is the same as the motion of the sidewalk)
The sidewalk's velocity = 1.5 m/s (since the sidewalk is moving at a speed of 1.5 m/s)
Therefore, the passenger's velocity relative to the stationary person alongside the moving sidewalk is 1.0 m/s + 1.5 m/s = 2.5 m/s.
b. To find the passenger's velocity relative to a person standing stationary on the moving sidewalk, we can subtract the velocities. Since they are moving in the same direction, we can simply subtract the magnitudes (taking into account the direction).
The passenger's velocity = 1.0 m/s (direction is the same as the motion of the sidewalk)
The sidewalk's velocity = 1.5 m/s (since the sidewalk is moving at a speed of 1.5 m/s)
Therefore, the passenger's velocity relative to the stationary person on the moving sidewalk is 1.0 m/s - 1.5 m/s = -0.5 m/s (negative sign indicates opposite direction).
c. To find the passenger's velocity relative to a person walking alongside the sidewalk with a speed of 2.0 m/s (opposite to the motion of the sidewalk), we can add the magnitudes of the velocities (taking into account the direction).
The passenger's velocity = 1.0 m/s (direction is the same as the motion of the sidewalk)
The sidewalk's velocity = 1.5 m/s (since the sidewalk is moving at a speed of 1.5 m/s)
The person's velocity walking alongside the sidewalk = -2.0 m/s (opposite to the direction the sidewalk is moving)
Therefore, the passenger's velocity relative to the person walking alongside the sidewalk is 1.0 m/s + 1.5 m/s + (-2.0 m/s) = 0.5 m/s (opposite to the direction the sidewalk is moving).
d. To find the passenger's velocity relative to a person riding in a cart alongside the sidewalk with a speed of 5.0 m/s (in the same direction as the sidewalk), we can subtract the magnitudes of the velocities (taking into account the direction).
The passenger's velocity = 1.0 m/s (direction is the same as the motion of the sidewalk)
The sidewalk's velocity = 1.5 m/s (since the sidewalk is moving at a speed of 1.5 m/s)
The person in the cart's velocity alongside the sidewalk = 5.0 m/s (in the same direction)
Therefore, the passenger's velocity relative to the person in the cart alongside the sidewalk is 1.0 m/s + 1.5 m/s - 5.0 m/s = -2.5 m/s (negative sign indicates opposite direction).
e. To find the passenger's velocity relative to a person riding in a cart with a speed of 4.0 m/s (perpendicular to the direction the sidewalk is moving), we can calculate the vector sum of the velocities.
For perpendicular vectors, we can use the Pythagorean theorem to find the magnitude:
The passenger's velocity = 1.0 m/s (direction is the same as the motion of the sidewalk)
The sidewalk's velocity = 1.5 m/s (since the sidewalk is moving at a speed of 1.5 m/s)
The person in the cart's velocity perpendicular to the sidewalk = 4.0 m/s (perpendicular to the direction the sidewalk is moving)
Using the Pythagorean theorem,
Magnitude of the passenger's velocity relative to the person in the cart = sqrt[(1.0 m/s)^2 + (1.5 m/s)^2]
= sqrt[1.0 m^2/s^2 + 2.25 m^2/s^2]
= sqrt[3.25 m^2/s^2]
= 1.80 m/s
Therefore, the passenger's velocity relative to the person in the cart riding perpendicular to the sidewalk is 1.80 m/s.
a. To find the time it takes for the passenger walking on the sidewalk to get from one end to the other, we can use the formula:
Time = Distance / Velocity
The distance of the sidewalk is given as 100.0 m and the passenger's velocity is given as 1.0 m/s.
Time = 100.0 m / 1.0 m/s
= 100.0 s
Therefore, it takes the passenger 100.0 seconds to walk from one end of the sidewalk to the other.
b. To find the time saved by taking the moving sidewalk instead of walking alongside it, we need to compare the time taken by the passenger on the moving sidewalk with the time it would take if the passenger walked alongside the stationary sidewalk.
As calculated in part (a), the time taken by the passenger while using the moving sidewalk is 100.0 seconds.
To find the time it would take if the passenger walked alongside the stationary sidewalk, we can use the formula:
Time = Distance / Velocity
The distance of the sidewalk is given as 100.0 m, and the passenger's walking speed is 1.0 m/s.
Time = 100.0 m / 1.0 m/s
= 100.0 s
Therefore, the passenger does not save any time by taking the moving sidewalk instead of walking alongside it.
To solve these questions, we need to understand the concept of relative velocity. Relative velocity is the velocity of an object as observed from a different reference point or observer.
First, let's calculate the passenger's velocity relative to the following observers:
a. To find the passenger's velocity relative to a person standing stationary alongside the moving sidewalk, we need to add the velocities. The passenger is walking at 1.0 m/s, and the sidewalk is moving at 1.5 m/s. Adding these velocities, we get:
Passenger's velocity = 1.0 m/s + 1.5 m/s = 2.5 m/s (with respect to the stationary person)
b. To find the passenger's velocity relative to a person standing stationary on the moving sidewalk, we subtract the sidewalk's velocity from the passenger's walking velocity:
Passenger's velocity = 1.0 m/s - 1.5 m/s = -0.5 m/s (with respect to the stationary person on the sidewalk)
Note: The negative sign indicates that the passenger is walking in the opposite direction to the motion of the moving sidewalk.
c. To find the passenger's velocity relative to a person walking alongside the sidewalk with a speed of 2.0 m/s in the opposite direction, we subtract the velocities:
Passenger's velocity = 1.0 m/s - 2.0 m/s = -1.0 m/s (with respect to the person walking).
Again, the negative sign indicates that the passenger is walking in the opposite direction.
d. To find the passenger's velocity relative to a person riding in a cart alongside the sidewalk with a speed of 5.0 m/s in the same direction, we subtract the velocities:
Passenger's velocity = 1.0 m/s - 5.0 m/s = -4.0 m/s (with respect to the person in the cart).
Similarly, the negative sign indicates that the passenger is walking in the opposite direction.
e. To find the passenger's velocity relative to a person riding in a cart with a speed of 4.0 m/s in a direction perpendicular to the sidewalk's motion, we use the Pythagorean theorem:
Passenger's velocity = √(1.0 m/s)^2 + (4.0 m/s)^2 = √(1.0 m/s)^2 + (16.0 m/s)^2 = √1.0 m^2/s^2 + 16.0 m^2/s^2 = √17.0 m^2/s^2 ≈ 4.12 m/s.
Now let's move on to the second part of the question:
a. To calculate the time it takes for the passenger to get from one end of the sidewalk to the other, we divide the length of the sidewalk (100.0 m) by the passenger's velocity (1.0 m/s):
Time = Distance / Velocity = 100.0 m / 1.0 m/s = 100.0 s.
Therefore, it takes the passenger 100.0 seconds to travel the length of the sidewalk.
b. To find how much time the passenger saves by taking the moving sidewalk instead of walking alongside it, we need to calculate the time it would have taken the passenger to walk the same distance (100.0 m) at their walking speed (1.0 m/s):
Time walking = Distance / Walking velocity = 100.0 m / 1.0 m/s = 100.0 s.
The time saved can be calculated by subtracting the time taken with the moving sidewalk from the time taken by walking:
Time saved = Time walking - Time with moving sidewalk = 100.0 s - 100.0 s = 0.0 s.
Therefore, the passenger does not save any time by taking the moving sidewalk compared to walking alongside it.
I hope this helps! Let me know if you have any further questions.