Two acceleration questions. Need answers with deductions!

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1. A body on a straight road with constant acceleration takes a 10 meter long track in 1,06 sec, and the next 10 m track in 2,2 sec. What is the acceleration?

2. A plane with constant acceleration until the half of the runway reaches the 80% of the velocity required for the take off.
What fractional part of the track does it reach the take-off speed?

1) You want to first find the average velocity during each segment. These average velocities are the the instantaneous velocities when the time is halfway through a segment ( 0.53 sec for v1, and 1.1 sec for v2). Using these we can find the acceleration:

$$d_1 = v_1 t_1$$
$$d_2=v_2 t_2$$
$$a = 2\frac{v_2-v_1}{t_2+t_1}$$

*Note that the 2 comes from the fact we want only HALF the time in each segment.

Being that the second segment takes longer than the first segment, you should expect your answer to be negative.

2) Assuming the acceleration is constant the entire time:

$$v_f^2=v_0^2+2a(x_f-x_0)$$

Simplifies to

$$(\frac{4}{5} v)^2 = 2a(\frac{1}{2}d)$$

which we can compare to an equation for the FULL take-off speed:

$$v^2 = 2a x_{takeoff}$$

The two equations can then be divided leaving

$$\frac{25}{16}= \frac{x_{takeoff}}{0.5 d}$$

Which results in

$$x_{takeoff}= \frac{ 25}{32} d$$
answered Nov 17, 2014 by (2,140 points)
edited Nov 17, 2014 by jcf1559
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