Question

A car starts at rest and moves along a perfectly straight highway with an acceleration of α1 = 10 m/s2 for a certain amount of time t1. It then moves with constant speed (zero acceleration) for a time t2 and finally decelerates with an acceleration α2= -10 m/s2 for a time t3 until it comes to a complete stop. The total time of motion is t1 +t2+t3=25 s. The total distance travelled by the car is 1 km. Find t2 Hints: (i) Recognize that each segment of the journey is at constant acceleration! (ii) What is the relationship between the quantities t1, t2, and t3? Use this to help simplify the set of equations that you obtain during the solution process

Answer 1

t1 = t3 = 5 seconds

t2 = 15 seconds

Step-by-step explanation:

For t = t1

a = 10 m/s^2

v(t) = 10*t

s(t) = 5*t^2

Distance traveled = 5*t1^2

For t = t2

a = 0 m/s^2

v(t) = 10*t1

s(t) = 10*t1*t

Distance traveled = 10*t1*t2

For t = t3

a = -10 m/s^2

v(t) = -10*t

s(t) = - 5t^2

Distance traveled = 5t3^2

Sum of all distances = 5*t1^2 + 10*t1*(t2) + 5t3^2

1000 = 5t1^2 + 10t1t2 + 5t3^2 + 10*t1*t2 ....... Eq 1

Distance traveled in first and last segments are the same:

t1 = t3 ..... Eq 2

Given: t1+t2+t3 = 25 .... Eq 3

Solving Equations simultaneously:

Subs Eq 2 into Eq 3 & Eq 1

1000 = 5t1^2 + 10*t1*t2 + 5*t1^2

100 = t1^2 + t1*t2 ..... Eq 4

2t1 + t2 = 25

t2 = 25 - 2t1 .... Eq 5

Subs Eq 5 into Eq 4

100 = t1^2 + t1*(25 - 2t1)

t1^2 -25t1 + 100 = 0

Solve for t1

t1 = 5 , 20 Hence, t1 = 5 sec is selected

t1 = t3 = 5 sec

t2 = 15 sec