Question 33
An aircraft is at rest at one end of
a runway which is 2.2 km long. The aircraft accelerates along the runway with
an acceleration of 2.5 m s-2 until it reaches its take-off speed of
75 m s-1. Calculate
(i) the time taken to reach take-off
speed
(ii) the distance travelled in this
time
(iii) Just as the aircraft reaches
take-off speed a warning light turns on. The maximum possible deceleration of
the aircraft is 4.0 m s-2 and 2.5 s elapses before the pilot takes
any action, during which times the aircraft continues at its take off speed.
Determine whether or not the aircraft can be brought to rest in the remaining
length of runway.
Reference: Cambridge International AS and A Level
Physics Coursebook – 2nd edition – by David Sang, Graham Jones,
Gurinder Chadha and Richards Woodside
Solution:
(i)
Initial speed, u = 0 m s-1
(aircraft is at rest)
Acceleration, a = 2.5 m s-2
Final speed, v = 75 m s-1
(take-off speed)
a = (v - u) / t
Time taken to reach
take-off speed, t = (v – u) / a = (75 – 0) / 2.5 = 30 s
(ii)
Consider the equation for
uniformly accelerated motion:
s = ut + ½ at2
Distance travelled during
this time, s = (0×30) + (½ ×2.5×302) = 1125 m
Alternatively,
Consider the equation for
uniformly accelerated motion:
v2 = u2
+ 2as
752 = 02
+ (2×2.5×s)
Distance travelled during
this time, s = 752 / (2×2.5) = 1125 m
(iii)
Remaining length of runway
when warning light turns on = 2200 – 1125 = 1075 m
Distance travelled during
the 2.5 s before the pilot takes any action:
Speed (= distance / time)
= 75 m s-1
Distance = Speed × Time =
75 × 2.5 = 187.5 m
Remaining length of runway
for deceleration = 1075 – 187.5 = 887.5 m
Consider the deceleration
to be maximum (= 4 m s-2) [acceleration, a = – 4 m s-2]
Initial speed, u = 75 m s-1
and final speed, v = 0 m s-1 (aircraft brought to rest)
Let’s try to calculate the
distance s required to bring the aircraft to rest and compare this with the
remaining length of runway.
Consider the equation for
uniformly accelerated motion:
v2 = u2
+ 2as
02 = 752
+ 2×-4×s
s = 752 / 8 =
703.125 m
Since the distance
required to bring the aircraft to rest (703.125 m) is less than the remaining
length of runway (887.5 m), the aircraft can be brought to rest in the
remaining length of runway.
