Question

To save fuel, some truck drivers try to maintain a constant speed when possible. A truck traveling at 29.0 km/hr approaches a car stopped at the red light. When the truck is 191.2 meters from the car the light turns green and the car immediately begins to accelerate at 2.3 m/s2. How close does the truck come to the car assuming the truck does not slow down? How far from the stop light has the car travelled when the truck reaches its closest distance?

Answer 1

Solution:-

29.0 km/hr * 1000 m/km * 1hr/3600 s = 8.05 m/s
The distance the truck moves = 8.05 * t

Let us assume the initial position of the truck = 0 m
Time versus position of the truck = 0 + 8.05 * t = 8.05 * t

The distance the car moves = ½ * 2.3 * t^2 = 1.15 * t^2
Let the initial position of the car = 191.2 m
The position of the car versus time = 191.2 + 1.15 * t^2

The distance between the truck and car versus time = position of the car versus time – position of the truck versus time

the distance between the truck and car = 191.2 + 1.15 * t^2 – 8.05 * t
Now we get a quadratic equation
distance = 1.15 * t^2 – 8.05 * t + 191.2

Now the graph is a parabola. The time when the slope of the parabola = 0 is the time when the truck is closest to the car

Slope = derivative

The derivative = 2 * 1.15 * t – 8.05

= 2.3 * t – 8.05
The derivative = 0

2.3 * t – 8.05 = 0
t = 8.05/2.3 = 3.5 seconds

distance = 1.15 * 3.5^2 – 8.05 * 3.5 + 191.2

distance = 177.11 m

The distance the car moves = ½ * 2.3 * t^2 = 1.15 * t^2
The distance the car moves = 1.15 * 3.5^2 = 14.09a m