Question

1. To save fuel, some truck drivers try to maintain a constant speed when possible. A truck traveling at 116.0 km/hr approaches a car stopped at the red light. When the truck is 177.4 meters from the car the light turns green and the car immediately begins to accelerate at 3.00 m/s^2 to a final speed of 120.00 km/hr. How close does the truck come to the car assuming the truck does not slow down?

2. How far from the stop light has the car travelled when the truck reaches its closest distance?

Answer 1

116 km/hr = 32.22 m/s

x truck = -177.4 + 32.22*t

xcar = 0 + 0*t + 0.5*3*t^2

so distance between car and truck is xcar - xtruck

xcar - xtruck = 1.5 t^2 - 32.22*t + 177.4

we want to find min so take derivative

d(xcar-xtruck)/dt = 3*t - 32.22

t = 32.22/3

so xcar - xtruck = 1.5*(32.22/3)^2 - 32.22*(32.22/3) + 177.4= 4.38 m

2) xcar = 0.5*3*(32.22/3)^2= 173 m

Answer 2

Answer 3

A truck traveling at 116.0 km/hr approaches a car stopped at the red light

velocity of truck=V_t=116.0 km/hr=32.22 m/s

initial distance between car and truck when green light is on

D_i=177.4 meters

initial speed of car=V_ic=0 (as the car started from rest)

final speed of car=V_fc=120.00 km/hr=33.33 m/s

acceleration of car=a_c=3.00 m/s²

from the kinematic equations

V_fc=V_ic+a_ct

33.33=0+3t

t=11.11s

In 11.11 seconds the car reaches to final speed

given that the truck does not slow down

after 11.11 sec both travel with their constant speeds which is not of our concern

our concern is that the distance has to be minimum

d/dt(D)=0

difference of velocities is zero can be taken

at given instance of time velocity of car=a_ct=3t

velocity of truck=V_t=32.22m/s

d/dt(D)=0 => 3t-32.22=0

3t=32.22

t=32.22/3=10.74 sec

after 10.74 sec they are close

in 10.74 sec Distance travelled by car=1/2 at²=1/2 x (3) x (10.74)²=173.02 m

Distance travelled by truck=V_t t=32.22 x 10.74=346.04 m

Difference of distances travelled = 346.04 - 173.02=173.02 m

Intial distance between the car and truck=177.4 m

Truck has covered 173.02 m

closest distance between them =177.4 - 173.02= 4.38 m

2. How far from the stop light has the car travelled when the truck reaches its closest distance?

It was already calculated

the closest distance is after 10.74 sec

In this time thae car has travelled

Distance travelled by car=1/2 at²=1/2 x (3) x (10.74)²=173.02 m