question 1

Information iOHO'O' 00 (50' ($901 The red car's front bumper is one car-length (which we will dene as L, and equals the length of each car and is one-quarter the length of the truck) behind a blue car's rear bumper, while a truck approaches in the other lane. To the drivers of these vehicles, the separation of the front bumper of the blue car and front bumper of the truck appears to be about twenty car-lengths. All three vehicles are traveling at the speed limit (which we will call \"a ). We will call the maximum acceleration possible for the cars a when speeding up, and b when braking. The driver of the red car is considering passing the blue car. Analysis 1. Ln We would like to express the motions of the cars and truck on a common footing, so we will dene the time of the situation described as t : 0. We will also choose the position of the rear bumper of the car at t : 0 to be the position a: : 0, with the positive direction being forward (we will not worry about the positions in other directions, like the separation of the lanes). With these denitions, we have the initial positions of the bumpers of all three vehicles: 0 red car rear bumper: a: : 0, front bumper: x : L 0 blue car rear bumper: a? : 2L, front bumper: :r = 3L 0 truck rear bumper: :1: = 27L, front bumper: :1: = 23L Given that all three vehicles are moving at the same constant speed \"0, we can write down their rather simple equations of motion using 55 (t) : vat + 550, with ma being dened above as the starting position. Note that the truck is moving in the opposite direction of the two cars, so the sign of its velocity is negative 0 red car rear bumper: 33w (t) : vat 'l' 0, front bumper: 55m\" (t) : "'ot + L 0 blue car rear bumper: 33hr (15) : \"at + 2L , front bumper: 55b)\" (t) : \"of + 3L 0 truck rear bumper: 1313:" (t) : ivot + 27L , front bumper: xtf (t) : _"'0t + 23L If the driver of the red car feels like the truck is too close to make the attempt, then she will have to wait for the truck to completely drive past before she pulls out, which means the rear of the truck must reach the rear of the red car. The rear bumpers of the red car and truck are aligned (same a: value) at a specic moment in time, so we can nd the time the driver will have to wait to pass by solving equations for their rear bumpers simultaneously: 27L 2:\": (t) : 37a (t) => not : foot + 27L :> t : E Ll] If the driver of the red car feels like she can pass the car before \"encountering\" the truck, she will obviously need to accelerate her car. Wanting to leave nothing to chance, she will obviously use the maximum acceleration a, and then the equations of motion for her car's bumpers are changed to those for constant linear acceleration: 1 t 1 :1: f. : 70% +1; 73 :I: t : 7mg + *0 15+ L red car rear bumper: Tr ( ) 2 O , front bumper: Tf ( ) 2 0 If the driver of the red car decides to \"go for it\The speed limit of the road is 13.3, and the driver of the red car decides to pass. The driver of the blue car is unaware that she is passing, and does not change his speed. If a "car length" is 4.12 meters, find the minimum acceleration (in ) the red car must have to successfully pass. [provide at least two decimal places]Information An object moves in one dimension with the speed that varies according to the given formula. Let's define positive velocity to mean the object is moving to the right, and negative velocity when the object is moving to the left. The quantities A and 7 are constants. v ( t) = >+7 ( 7Analysis 1. D) U] The velocity is not linear but quadratic as a function of time. This means that the acceleration will not be constant, and kinematic equations for constant acceleration will not apply for this motion. We can solve for acceleration as a function of time by taking the derivative of the velocity function: (it! 27 : = 2 t dt 15 O (l Acceleration is positive if \"I > 0 and negative. if '7 U ). If both are negative, the object is moving to the left and speeding up (a 0 and \"I 0, then the object starts moving to the left and slows down until it turns around at the time found above and starts moving to the right and speeding up. We can also calculate the position as a function of time by integrating the velocity function: 7? 2 , $(t)=f-vdt=/[A+7'(t) ]dt=aro+x\\t+ (.g12)td O The object moving with the velocity described by the function does so from t = 0 tot = to, and starts with an initial velocity of 19.7%. Find the value of P}! such that the object's average velocity for this time period is zero. Indicate the sign of 7 using a + or Sign. [provide at least one decimal place]