A cart is driven by a large propeller or fan, which can accelerate or decelerate the cart. The cart starts out at the position x = 0 m, with an initial velocity of + 3.3 m/s and a constant acceleration due to the fan. The direction to the right is positive. The cart reaches a maximum position of x = + 11.6 m, where it begins to travel in the negative direction. Find the acceleration of the cart.
To find the acceleration of the cart, we can use the kinematic equation:
𝑥 = 𝑥0 + 𝑣0𝑡 + 0.5𝑎𝑡^2
where 𝑥 is the final position, 𝑥0 is the initial position, 𝑣0 is the initial velocity, 𝑎 is the acceleration, and 𝑡 is the time.
Given that 𝑥0 = 0 m, 𝑥 = +11.6 m, 𝑣0 = +3.3 m/s, and the cart starts at rest, we can rewrite the equation as:
11.6 = 0 + (3.3)𝑡 + 0.5𝑎𝑡^2
Simplifying the equation further, we get:
11.6 = 3.3𝑡 + 0.5𝑎𝑡^2
Now, since the cart is decelerating, the acceleration 𝑎 will be negative. Let's also assume that the time taken to reach the maximum position is 𝑡.
To find 𝑎, we need another equation involving 𝑡. We can use the equation for velocity:
𝑣 = 𝑣0 + 𝑎𝑡
At the maximum position, the velocity of the cart becomes zero. So we can write:
0 = 3.3 + 𝑎𝑡
Rearranging the equation, we have:
−3.3 = 𝑎𝑡
Now we can substitute this value of 𝑡 into the first equation:
11.6 = 3.3𝑡 + 0.5𝑎𝑡^2
Plugging in −3.3 for 𝑡, we get:
11.6 = (3.3)(−3.3) + 0.5𝑎(−3.3)^2
Simplifying further:
11.6 = −10.89 − 5.39𝑎
Rearranging the equation:
5.39𝑎 = −10.89 − 11.6
5.39𝑎 = −22.49
𝑎 = −22.49/5.39 ≈ -4.17 m/s^2
So, the acceleration of the cart is approximately -4.17 m/s^2.