Riding up an escalator while staying on the same step for the whole ride takes 29 s . Walking up the same escalator takes 20 s. If you are walking against the escalator and you are at the top of the escalator right now, how many seconds ago were you at the bottom? Express your answer with the appropriate units.
Riding up an escalator while staying on the same step for the whole ride takes 29 s . Walking up the same escalator takes 20 s.
In this problem, we use kinematic equations to model the motion of a person walking against an escalator. We then use the times it takes to ride up the escalator without walking and to walk up the escalator to solve for the time it takes for the person to travel the length of the escalator without walking. Finally, we subtract this time from the time it takes for the person to walk up the escalator to determine how long ago the person was at the bottom of the escalator.
Let's assume that the escalator has a certain speed, and let's denote this speed by v_e. Similarly, let's denote your walking speed by v_w.
When you are riding up the escalator while staying on the same step, your velocity relative to the ground is the sum of the escalator velocity and your stationary velocity, which is zero. Therefore, your velocity relative to the ground is simply the velocity of the escalator: v_e.
When you are walking up the same escalator, your velocity relative to the ground is the sum of your walking velocity and the velocity of the escalator. Therefore, your velocity relative to the ground is v_e + v_w.
We can use the following formula to calculate the length of the escalator:
Length of escalator = time taken × velocity of the escalator
Using this formula, we can write two equations for the length of the escalator, one for riding up the escalator and one for walking up the escalator:
For riding up the escalator:
L = 29v_e
For walking up the escalator:
L = 20(v_e + v_w)
Since you are walking against the escalator, your velocity relative to the ground is v_e - v_w. Therefore, we can use the following formula to calculate the time taken to travel the length of the escalator when walking against the escalator:
Time taken = Length of escalator ÷ velocity relative to the ground
Using this formula and substituting the values of L and v_e - v_w from the above equations, we get:
Time taken = L / (v_e - v_w) = (20(v_e + v_w)) / (v_e - v_w)
Now we can consider that the length of the escalator is always constant,
20(v_e + v_w) = 29v_e
Simplifying this equation, we get:
v_e = 20v_w / 9
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