A car enters a curved road (in the form of a quarter circle) of length 360m at a speed of 4kmph and leaves at 60kmph. If the car is travelling with constant acceleration along the curve, determine the magnitude and direction of acceleration when the car enters leaves the curve.
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Given info : A car enters a curved road (in the form of a quarter circle) of length 360m at a speed of 4kmph and leaves at 60kmph. the car is travelling with constant acceleration along the curve.
To find : the magnitude and direction of acceleration when the car leaves the curves.
solution : length of curve = 360 m
⇒2πr/4 = 360 m
⇒πr = 720
⇒r = 720/π m
let's find tangential acceleration, a_t
using formula, v² = u² + 2a_t s
here v = 60 km/h = 60 × 5/18 = 50/3 m/s
u = 4 km/h = 4 × 5/18 = 10/9 m/s
s = 360 m
so, (50/3)² = (10/9)³ + 2 × a_t × 360
⇒2500/9 = 100/81 + 720 × a_t
⇒(22500 - 100)/81 = 720 × a_t
⇒a_t = 22400/(81 × 720) ≈ 0.4 m/s²
now find angular acceleration, a_c
using formula, a_c = v²/r
= (50/3)²/(720/π)
= (2500/9)/(720/π)
= 1.2 m/s²
net acceleration, a = √{(a_c)² + (a_t)²}
= √{(1.2)² + (0.4)²}
= √(1.44 + 0.16)
≈ 1.265 m/s²
direction of acceleration , θ = tan¯¹ (a_t/a_c)
= tan¯¹ (0.4/1.2)
= tan¯¹(1/3)
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