wo fan carts with different fan speeds and different accelerations are started from rest some
distance from one another.
Cart A begins at position 0 with a velocity of zero and an acceleration of aA to the right.
Cart B begins at position xB with a velocity of zero and an acceleration of aB to the left.
1. Draw a qualitative position-time graph to represent the situation.
2. Develop an algebraic expression in terms of the variables below that will predict exactly where the carts will meet, indicated by xf.
please give a detailed answer with some explanation
To solve this problem, let's break it down step by step:
Step 1: Qualitative Position-Time Graph
To draw a qualitative position-time graph, we need to analyze the motion of both carts and determine how their positions change with time.
- Cart A:
Since Cart A has an acceleration to the right and starts from rest, its velocity will increase over time, and therefore, its position will also increase. The graph for Cart A would show a steadily increasing position with time.
- Cart B:
Cart B has an acceleration to the left. Since it starts from rest, its velocity will decrease over time, causing its position to decrease as well. The graph for Cart B would show a slowly decreasing position with time.
The two carts will eventually meet at a certain point on the graph. Let's label this point xf.
Step 2: Algebraic Expression for xf
To determine the exact value of xf, we need to find the time it takes for each cart to reach point xf.
- Cart A:
We can use the equation of motion, xA = vA₀t + (1/2)aA t², where xA is the position of Cart A, vA₀ is the initial velocity (zero in this case), aA is the acceleration of Cart A, and t is the time taken. Since Cart A starts from rest, the equation simplifies to xA = (1/2)aA t².
- Cart B:
Using the same equation of motion, xB = vB₀t + (1/2)aB t², where xB is the position of Cart B, vB₀ is the initial velocity (also zero), aB is the acceleration of Cart B, and t is the time taken. The equation simplifies to xB = -(1/2)aB t² since Cart B has an acceleration in the opposite direction.
For both carts to meet at xf, their positions at time t should be equal:
(1/2)aA t² = -(1/2)aB t²
To find xf, we can solve this equation for t:
(1/2)aA t² + (1/2)aB t² = 0
([aA + aB]/2) t² = 0
aA + aB = 0 (since t ≠ 0)
Solving for t:
t = 0 (not valid as it corresponds to the starting point)
aA + aB = 0
aA = -aB
Since aA is the acceleration of Cart A, and aB is the acceleration of Cart B, for the carts to meet, their accelerations must be equal in magnitude but opposite in direction.
Therefore, the exact position where the carts will meet, xf, is any point between the initial positions of Cart A and Cart B (i.e., xB and 0), depending on the specific values of aA and aB.