Two 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.
Xa = 0 + 0 t + .5 A t^2
Xb = XB + 0t - .5 B t^2
they will hit when Xa = Xb
.5 A t^2 = XB - .5 B t^2
.5 t^2 ( A+B ) = XB
t^2 = 2 XB/(A+B)
so
Xa = Xb = .5 A t^2 = A XB /(A+B)
That is the answer?
To answer the first question and draw a qualitative position-time graph, we need to understand the motion of the two carts.
- Cart A: It starts at position 0 with a velocity of zero and an acceleration of aA to the right. Since it has a positive acceleration, it will gradually increase its velocity and move to the right. The position-time graph for Cart A would be a straight line sloping upwards from left to right.
- Cart B: It starts at position xB with a velocity of zero and an acceleration of aB to the left. Since it has a negative acceleration, it will decelerate (reduce velocity) and move to the left. The position-time graph for Cart B would be a straight line sloping downwards from left to right.
Now let's move on to the second question and develop an algebraic expression to predict where the carts will meet.
Let's assume that Cart A and Cart B meet at position xf at time tf.
- For Cart A:
The position of Cart A can be represented as: xA = 0 + (1/2)aA(tf)^2 (using the equation of motion for constant acceleration).
- For Cart B:
The position of Cart B is given by: xB = xB + (1/2)aB(tf)^2.
Since the carts meet at position xf, we can equate the positions of Cart A and Cart B:
0 + (1/2)aA(tf)^2 = xB + (1/2)aB(tf)^2.
Simplifying the equation gives:
(1/2)aA(tf)^2 - (1/2)aB(tf)^2 = xB.
To find the value of tf and xf, we need to know the specific values of aA, aB, and xB. By substituting these values into the equation, we can solve for tf and calculate xf.