Question

2 motion with constant acceleration (continued)
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1. displacement given initial and final velocities and the constant acceleration of an object, what mathematical relationship would you use to find the displacement?
2. acceleration a woman driving west along a straight road at a speed of 23 m/s sees a deer on the road ahead. she applies the brakes when she is 210 m from the deer. if the deer does not move and the car stops right before it hits the deer, what is the acceleration provided by the cars brakes?
3. distance the airplane in figure 18 starts from rest and accelerates east at a constant 3.00 m/s² for 30.0 s before leaving the ground.

a. what was the planes displacement (δx)?
b. how fast was the airplane going when it took off?

Explanation:

Step1: Recall the kinematic - equation

The kinematic equation $v^{2}=v_{0}^{2}+2a\Delta x$ can be used to find displacement $\Delta x$ when initial velocity $v_{0}$, final velocity $v$, and acceleration $a$ are known. We can re - arrange it as $\Delta x=\frac{v^{2}-v_{0}^{2}}{2a}$.

Step2: Identify given values for problem 34

Initial velocity $v_{0}=23\ m/s$, final velocity $v = 0\ m/s$, and displacement $\Delta x=210\ m$.

Step3: Use the kinematic equation for acceleration

From $v^{2}=v_{0}^{2}+2a\Delta x$, we can solve for $a$. Rearranging gives $a=\frac{v^{2}-v_{0}^{2}}{2\Delta x}$. Substituting $v = 0$, $v_{0}=23\ m/s$, and $\Delta x = 210\ m$, we have $a=\frac{0-(23)^{2}}{2\times210}=\frac{- 529}{420}\approx - 1.26\ m/s^{2}$.

Step4: Identify values for problem 35a

Initial velocity $v_{0}=0\ m/s$, acceleration $a = 3.00\ m/s^{2}$, and time $t = 30.0\ s$. We use the equation $\Delta x=v_{0}t+\frac{1}{2}at^{2}$. Since $v_{0}=0$, $\Delta x=\frac{1}{2}\times3\times(30)^{2}=1350\ m$.

Step5: Identify values for problem 35b

We use the equation $v=v_{0}+at$. With $v_{0}=0\ m/s$, $a = 3.00\ m/s^{2}$, and $t = 30.0\ s$, we get $v=0 + 3\times30=90\ m/s$.

Answer:

1. $\Delta x=\frac{v^{2}-v_{0}^{2}}{2a}$
2. $a\approx - 1.26\ m/s^{2}$
3. a. $\Delta x = 1350\ m$

b. $v = 90\ m/s$