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The driver of a car travelling at 30 metres per second has a reaction time of 1.5 seconds.
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The car’s brakes decelerate the car at 3.75 metres per second squared once they are activated.
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How much time does the car take to stop, including the driver’s thinking time?
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Okay, so, in this question, we’ve been told that we’re dealing with a car.
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So, let’s say this block is going to represent our car.
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And we’ve been told that it’s travelling at 30 metres per second initially.
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So, let’s assume it’s initially travelling towards the right at 30 metres per second.
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Now we’ve been told that the driver has a reaction time of 1.5 seconds.
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This means that if the driver notices a hazard at this point, then the car continues to move for another 1.5 seconds before the driver can actually press the brakes.
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And this is because the driver’s reaction time is 1.5 seconds.
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And therefore, it took them that long to react to the hazard that they noticed 1.5 seconds earlier.
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So, until the brakes are pressed, the car is still travelling towards the right at 30 metres per second.
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And we can say that the time interval here is 1.5 seconds as we’ve been told in the question
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Now after the driver’s pressed the brakes, we know that the car will decelerate.
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And at some point in the future, the car will eventually stop.
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So, we can say that at this position the car speed is zero metres per second.
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And as well as this, we know that once the driver presses the brakes, the car will decelerate at 3.75 metres per second squared.
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In other words, we can say that just after the brakes are pressed, the car’s acceleration becomes negative 3.75 metres per second squared.
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And the reason that it’s negative is because the car is decelerating.
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In other words, it’s slowing down.
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Now what we’ve been asked to do is to find the amount of time taken for the car to stop.
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And this is including the driver’s thinking time.
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Now the driver’s thinking time is simply the reaction time of 1.5 seconds.
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So, we need to find the total time taken from when the driver notices the hazard to when the car has stopped.
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And so, in order to do that, because we already know that this amount of time is 1.5 seconds, we need to go about finding this time interval.
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So, let’s say that the time taken between when the brakes are pressed and when the car actually stops is the time interval 𝑡.
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And during this entire time interval, the car has an acceleration of negative 3.75 metres per second squared because, remember, this time interval starts when the brakes are pressed.
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And as well as this, we know the initial velocity of the car, which is 30 metres per second, and the final velocity of the car, which is zero metres per second.
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So, based on all the information we know, we need to find an equation that links all of that information in order to find the time 𝑡.
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And the equation that we’re looking for is the definition of acceleration.
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We can recall that acceleration is defined as the change in velocity of an object divided by the time taken for that change in velocity to occur.
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Now in this case, we know the acceleration of the car, which is 𝑎, over the period of time, which is 𝑡.
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And as well as this, we know the initial and final velocities.
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So, we can use this information to calculate the value of 𝑡.
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We can firstly recall that the change in velocity is found by subtracting the initial velocity from the final velocity.
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And now what we can do is to rearrange this equation to solve for 𝑡.
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To do this, we multiply both sides of the equation by 𝑡 over 𝑎.
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This way, on the left-hand side the acceleration cancels, and on the right-hand side the time 𝑡 cancels.
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And what we’re left with is that the time 𝑡 is equal to the final velocity minus the initial velocity divided by the acceleration 𝑎.
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Then we can substitute in all our values.
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The final velocity of the car is zero metres per second.
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The initial velocity of the car is 30 metres per second.
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And the acceleration is negative 3.75 metres per second squared.
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And now we can see that because we’ve taken care to put a negative sign in front of the acceleration, that our value for time is going to be positive because we’ve got zero minus a positive value.
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And so, the numerator is going to be negative and the denominator is already negative.
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Now a negative value divided by a negative value gives a positive value.
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And when we evaluate the right-hand side, we find that this time interval 𝑡 is eight seconds.
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So, we found the time taken between when the brakes are pressed and when the car actually stops.
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That time interval is eight seconds.
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And now since we’ve been asked to find the time taken for the car to stop including the driver’s thinking time, all we need to do is to add up this time interval to this time interval.
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And so, we can say that the total time taken for the car to stop, which we’ll call 𝑡 subscript tot, is equal to the first time interval, which is 1.5 seconds, plus the second time interval, which is eight seconds, which is the amount of time the car takes to stop once the brakes have been pressed.
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And once we evaluate the right-hand side of the equation, we’ve found the answer to our question.
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The time taken for the car to stop, including the driver’s thinking time, is 9.5 seconds.