During a long airport layover, a physicist father and his 8-year-old daughter try a game that involves a moving walkway. They have measured the walkway to be 41.5 m long. The father has a stopwatch and times his daughter. First, the daughter walks with a constant speed in the same direction as the conveyor. It takes 15.1 s to reach the end of the walkway. Then, she turns around and walks as before with the same speed relative to the conveyor, just this time in the opposite direction. The return leg takes 72.1 s.
a) What is the speed of the walkway conveyor relative to the terminal?
b) With what speed was the girl walking?
To solve this problem, we can set up an equation using the distance formula.
Let's first define the variables:
- v_w: speed of the walkway conveyor (in m/s)
- v_g: speed of the girl relative to the ground (in m/s)
a) To find the speed of the walkway conveyor relative to the terminal, we need to determine the time it takes for the girl to walk without the walkway, and then find the difference between the time with and without the walkway.
The daughter walks with a constant speed in the same direction as the conveyor. It takes 15.1 s to reach the end of the walkway. Since she walks with the same speed as the conveyor, her speed relative to the ground (speed of the girl without the walkway) would be v_g - v_w.
The equation for the first part of the trip is:
Distance = Speed x Time
41.5 m = (v_g - v_w) x 15.1 s
b) Now, when the daughter turns around and walks in the opposite direction, she is walking against the conveyor. The daughter walks with the same speed relative to the conveyor, v_g, but this time it takes her 72.1 s to reach the end of the walkway.
The equation for the second part of the trip is:
Distance = Speed x Time
41.5 m = (v_g + v_w) x 72.1 s
To find the values of v_w and v_g, we need to solve these two equations simultaneously.
Let's solve them now:
First equation: 41.5 = (v_g - v_w) x 15.1
Rearranging the equation: v_g - v_w = 41.5 / 15.1
Second equation: 41.5 = (v_g + v_w) x 72.1
Rearranging the equation: v_g + v_w = 41.5 / 72.1
Now, add the two equations together:
(v_g - v_w) + (v_g + v_w) = (41.5 / 15.1) + (41.5 / 72.1)
Simplifying the equation:
2v_g = (41.5 / 15.1) + (41.5 / 72.1)
Dividing both sides by 2:
v_g = [(41.5 / 15.1) + (41.5 / 72.1)] / 2
Now we can plug in the values and calculate:
v_g = [(41.5 / 15.1) + (41.5 / 72.1)] / 2
v_g ≈ 1.930 m/s
To find the speed of the walkway conveyor relative to the terminal (v_w), we can substitute v_g into the first equation:
v_g - v_w = 41.5 / 15.1
Rearranging the equation to solve for v_w:
v_w = v_g - (41.5 / 15.1)
Substituting the value of v_g into the equation:
v_w = 1.930 - (41.5 / 15.1)
v_w ≈ -0.303 m/s (or -0.303 m/s relative to the terminal)
Since the conveyor is moving in the opposite direction of the girl, the speed of the walkway conveyor relative to the terminal is approximately 0.303 m/s.
The speed of the girl walking is approximately 1.930 m/s.
To solve this problem, we need to break it down into two parts: the daughter's speed relative to the walkway, and the walkway's speed relative to the terminal.
a) To find the speed of the walkway conveyor relative to the terminal, let's consider the daughter's motion when she walks in the same direction as the conveyor. We can assume that her speed relative to the walkway is v (unknown), and the conveyor's speed is w (unknown to be determined). So, her total speed relative to the terminal will be v + w.
During the first leg of the journey, the daughter walks with a constant speed for 15.1 seconds and covers a distance equal to the length of the walkway, which is 41.5 m.
Using the equation: distance = speed × time, we have:
41.5 m = (v + w) × 15.1 s
Now, let's consider the daughter's motion when she walks in the opposite direction to the conveyor. Her speed relative to the walkway is still v, but the conveyor's speed is now -w (opposite direction). So, her total speed relative to the terminal will be v - w.
During the return leg, the daughter walks for 72.1 seconds and covers a distance of 41.5 m.
Using the equation: distance = speed × time, we have:
41.5 m = (v - w) × 72.1 s
Now, we have a system of two equations with two unknowns (v and w). We can solve these equations simultaneously to find the values of v and w.
b) To find the daughter's speed relative to the walkway, we can use either of the two equations above. Let's use the first equation:
41.5 m = (v + w) × 15.1 s
Now, let's solve these equations to find the values of v and w.