Question

A car is driving at a speed of 59.0 km/h toward an intersection just as the light changes from green to yellow. The driver has a reaction time of 0.750 s and the magnitude of the braking acceleration of the car is 5.50 m/s2. Find the minimum distance xmin the car travels before coming to a stop after the light changes. (Express to three sig figs)

Answer 1

The minimum distance the car travels before coming to a stop is 1225 m.

Step-by-step explanation:

To find the minimum distance the car travels before coming to a stop, we need to consider the distance traveled during the driver's reaction time and the distance traveled while the car is decelerating.

First, we calculate the distance traveled during the reaction time using the equation:

distance = speed × time

distance = 59.0 km/h × 0.750 s = 44.3 m

Next, we calculate the distance traveled while decelerating using the equation:

distance = (initial velocity² - final velocity²) / (2 × acceleration)

distance = (59.0 km/h)² / (2 × 5.50 m/s²) = 1181 m

The minimum distance the car travels before coming to a stop is the sum of the distances traveled during the reaction time and while decelerating:

minimum distance = 44.3 m + 1181 m = 1225 m

Answer 2

The car travels 36.8 m before coming to stop after the light changes

Step-by-step explanation:

The car moves at a constant speed of 59.0 km/h for 0.750 s before the driver hits the brake.

The equation for the position of an object moving at constant speed is:

x = x0 + v t

where:

x = position at time t

x0 = initial position

v = speed

t= time

Let´s consider the initial position as the position at which the driver sees the traffic light and decides to brake. That will make x0 = 0. Then, the position after 0.750 s will be:

x = 59.0 km/h * 0.750 s (1 h /3600 s) = 0.0123 km (1000 m / 1 km) = 12.3 m

while braking, the car has a negative acceleration, then, the speed is not constant. The position of the car will be given by the following equation:

x = x0 + v0 t + 1/2 a t² ( where a = acceleration and v0 = initial speed)

and the speed can be expressed as follows:

v = v0 + a t

from this equation, we can calculate how much time it takes the car to stop (v = 0):

0 = v0 + a t

-v0 = a t

-v0 / a = t

v0 is the speed of the car as the driver hits the brake (59.0 km/h) and "a" is the acceleration (5.50 m/s²) that will be negative because the car is losing speed. Then:

-59.0 km/h (1000 m / 1 km) (1 h / 3600 s) / (-5.50 m/s²) = 2.98 s

Now, we can calculate the position at this time to know the minimum distance the car travels before coming to stop:

x = x0 + v0 t + 1/2 a t²

now x0 will be the distance traveled after the driver sees the light but before braking ( 12.3 m). v0 will be the speed before braking, 59.0 km / h or 16.4 m/s. Then:

x = 12.3 m + 16.4 m/s * 2.98 s +1/2 (-5.50 m/s²) * (2.98 s)²

x = 36.8 m