Suppose a car is traveling at 14.0 m/s, and the driver sees a traffic light turn red. After 0.500 s has elapsed (the reaction time), the driver applies the brakes, and the car decelerates at 5.00 m/s2. What is the stopping distance of the car, as measured from the point where the driver first notices the red light?
To determine the stopping distance of the car, we can use the equations of motion.
First, let's calculate the distance covered by the car during the reaction time. The equation we'll use is:
\[ d_{\text{reaction}} = v_0 \cdot t_{\text{reaction}}\]
where
\(d_{\text{reaction}}\) is the distance covered during the reaction time,
\(v_0\) is the initial velocity of the car, and
\(t_{\text{reaction}}\) is the reaction time.
Plugging in the values:
\(v_0 = 14.0 \, \text{m/s}\) (initial velocity of the car) and
\(t_{\text{reaction}} = 0.500 \, \text{s}\) (reaction time of the driver),
we can calculate the distance covered during the reaction time:
\(d_{\text{reaction}} = 14.0 \, \text{m/s} \cdot 0.500 \, \text{s}\).
Now, let's calculate the distance covered by the car during the deceleration phase. The equation we'll use is:
\[ d_{\text{deceleration}} = \frac{1}{2} \cdot a \cdot t_{\text{deceleration}}^2 \]
where
\( d_{\text{deceleration}} \) is the distance covered during the deceleration phase,
\( a \) is the deceleration (negative acceleration) of the car, and
\( t_{\text{deceleration}} \) is the time taken by the car to decelerate from its initial velocity to zero.
We are given \( a = -5.00 \, \text{m/s}^2 \) (deceleration) and
we need to find \( t_{\text{deceleration}} \).
The formula to find \( t_{\text{deceleration}} \) is:
\[
t_{\text{deceleration}} = \frac{{v_f - v_0}}{{a}}
\]
where
\( v_f \) is the final velocity of the car, and
\( v_0 \) is the initial velocity of the car.
Since the car comes to a stop, \( v_f \) is 0, so we have:
\[
t_{\text{deceleration}} = \frac{{0 - 14.0}}{{-5.00}}
\]
Finally, we can calculate the distance covered during the deceleration phase:
\[
d_{\text{deceleration}} = \frac{1}{2} \cdot -5.00 \cdot \left( \frac{{0 - 14.0}}{{-5.00}} \right)^2
\]
To find the total stopping distance, we just need to add the values of \( d_{\text{reaction}} \) and \( d_{\text{deceleration}} \):
\[
\text{Stopping distance} = d_{\text{reaction}} + d_{\text{deceleration}}
\]