GA4_solns.pdf - PHYS 2210 Spring 2014 GA4 Solutions Problem...

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PHYS 2210 — Spring 2014 GA4 Solutions Problem 1 Force and Simple Dynamics: Exit Ramp On a trip through Parley’s Canyon, you notice that when the freeway goes steeply down a hill, there are emergency exits every few miles. These emergency exits are straight ramps which leave the freeway and are sloped uphill. They are designed to stop runaway trucks and cars that lose their brakes on downhill stretches of the freeway even if the road is covered with ice. You are curious, so you stop at the next emergency exit to take some measurements. You determine that the exit rises at an angle of 10 from the horizontal and is 100 m long. What is the maximum speed of a truck that you are sure will be stopped by this road, even if the frictional force of the road surface is negligible? Solution Conceptual Analysis The maximum speed that the truck can have and still be stopped by the 100 m road is the speed that it can go and be stopped at exactly 100 m. Since there is no friction, this problem is similar to a projectile problem: You can think of the problem as being a ball tossed into the air except here you know the highest point and you are looking for the initial velocity needed to reach that point. Because there is an incline, the value of the acceleration due to gravity is not simply g ; it is the component of gravity acting parallel to the incline. Strategic Analysis Calculate the projection (component) of g along the inclined surface, g k . This is the acceler- ation that the truck will experience as it travels along the ramp’s length. Use the 1D kinematic equations for constant acceleration to find the maximum initial speed of the truck. Quantitative Analysis First, label the important quantities: θ = 10 - incline of the ramp relative to the horizontal g k = g sin θ - component of gravitational acceleration along the inclined surface v i - initial velocity of the truck at the bottom of the ramp (what is being sought) 1
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2 v f - final velocity of the truck at the top of the ramp ( v f = 0 ) L = 100 m - length of ramp The most straightforward way to proceed is just to use the 1D kinematic equation for constant acceleration in which the time has eliminated: x f - x i = v 2 f - v 2 i 2 a
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