A car drives along a road at 15 meters per second toward a bridge, as shown in the diagram. When the front wheels of the car are 50 meters from the bridge, the driver sees a sign warning that the bridge has a collapsed section. The car can decelerate at five meters per second squared. What is the maximum reaction time that the driver can have and still stop the car before it reaches the bridge? Answer to one decimal place.
Our diagram shows us that for this car traveling along at 15 meters per second, when the car is 50 meters away from a bridge with a collapsed portion, the driver notices a sign saying so. The time between when the driver first sees the sign and when the driver presses down on the brake pedal to begin deceleration is called the reaction time of the driver. In this question, we want to know how long can the driver’s reaction time be and the car still stop before it reaches the bridge.
To begin figuring this out, let’s clear some space on screen and recall that, in general, the total stopping distance of a vehicle is equal to what’s called the thinking distance plus the braking distance. Thinking distance, as shown in our diagram, is the distance the vehicle travels while the driver is reacting to the situation and getting to the point of braking. Only when the brake is pressed does the vehicle begin to decelerate. And the distance it travels while doing so is called the braking distance. At the end of the braking distance, the vehicle has a speed of zero; it’s come to rest.
In this situation, we’re told that the total stopping distance of our vehicle is 50 meters. That’s how far it can travel before reaching the bridge. So 50 meters is equal to the thinking distance of this vehicle plus the braking distance. Recalling that, in general, speed 𝑣 is equal to distance traveled 𝑑 divided by the time taken to travel that distance 𝑡, we can slightly rearrange this equation by multiplying both sides by the time 𝑡 so that we find that distance equal speed times time. That tells us that when it comes to the thinking distance of our vehicle, that will be equal to the initial speed of the vehicle, that’s 15 meters per second, multiplied by the time it takes the driver to react to the situation of the bridge being out. We’ll call that the reaction time, 𝑡 sub r.
15 meters per second times 𝑡 sub r is our thinking distance. And when it comes to our braking distance, we know that this involves our car experiencing a constant deceleration of five meters per second squared. Because this acceleration is constant, the car’s motion is described by the kinematic equations of motion. Specifically it’s described by the equation final velocity squared equals initial velocity squared plus two times acceleration multiplied by displacement 𝑠.
For our object undergoing constant acceleration — that is, our vehicle — its final velocity is zero. Its initial velocity 𝑣 sub i is 15 meters per second to the right. And if we say that motion to the right is positive, that means that our acceleration 𝑎, which points to the left, must be negative. This means that, for us, two times 𝑎 times 𝑠 is equal to two times negative five meters per second squared times 𝑠, where 𝑠 is the braking distance of our vehicle.
Our equation of motion says that if we add this term to 15 meters per second squared, we get zero. Rewriting this equation where we have some space to work, what we want to do here is solve for 𝑠, which is not a unit of seconds but is rather our braking distance in meters. To begin doing that, we subtract 15 meters per second squared from both sides. That way, 15 meters per second squared cancels out with negative 15 meters per second squared. Next, we divide both sides by two times our acceleration of negative five meters per second squared. That causes both the factor of two and the acceleration to cancel out on the right. That gives us an equation for the braking distance 𝑠, where 𝑠 is the subject.
Before we calculate 𝑠, notice that the negative sign in numerator and denominator cancel out. And then, since 15 squared is 225, our numerator becomes 225 meter squared per second squared and our denominator is 10 meters per second squared. Both the numerator and denominator have units of meters per second squared. So those cancel out, and we’re left just with units of meters. The braking distance 𝑠 of our vehicle is exactly 22.5 meters. We can substitute this value into our larger equation. And now, that equation reads the stopping distance of 50 meters equals the thinking distance of 15 meters per second times the reaction time 𝑡 sub r plus the braking distance of 22.5 meters.
Notice that since we’ve assumed a stopping distance of 50 meters, the maximum distance our vehicle can safely travel, we’re indeed solving for the maximum reaction time allowable, 𝑡 sub r. If we subtract 22.5 meters from both sides, then on the right-hand side, 22.5 meters cancels out entirely. And then dividing both sides by 15 meters per second, that speed cancels out on the right. That leaves us an expression for the reaction time 𝑡 sub r, where 𝑡 sub r is the subject.
In the numerator of this fraction, we have 50 meters minus 22.5 meters. That’s equal to 27.5 meters. When we calculate this fraction, we find the reaction time of 1.83 repeating seconds. However, we want to give our final answer to one decimal place. Rounding to one decimal place gives us 1.8 seconds. This is the maximum reaction time for the driver of this vehicle to bring it safely to rest.