2. A car weighing 2.5 metric tons and traveling at 90 km/h hits a 500 m...

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2. A car weighing 2.5 metric tons and traveling at 90 km/h hits a 500 m long stretch of black ice. Unfortunately, due to skidding, neither accelerating nor braking has any effect on the speed! The driver manages to maintain steady straight direction of motion and the only impact is provided by the ice friction force, which is numerically equal to 4v2 Newtons, where the velocity v of the car is measured in m/sec. (a) Using Newton's Second Law F = ma, set up a mathematical model for the position x(t) and velocity v(t) of the car as functions of time t. Start by drawing a diagram and choosing a consistent system of units based on kg, m, sec (1 ton = 1000 kg, 1 m/sec 3.6 km/h, 1 N = 1 kg m/sec). Introduce and label the variables, show the units and write down the differential equations and the intial conditions. = (b) Use the model in part a to calculate v(t) and x(t). Fully show the process of solving the initial value problems. (c) Based on your work so far, how long will it take to pass the iced over stretch of the road and how fast will the car be moving as it exits the ice? Do not use a calculator! Leave your answers in an "algebraic" form, which may involve exponential functions and logarithms, simplify as appropriate, and clearly specify the units (e.g. sec, m/sec or km/h). 2. A car weighing 2.5 metric tons and traveling at 90 km/h hits a 500 m long stretch of black ice. Unfortunately, due to skidding, neither accelerating nor braking has any effect on the speed! The driver manages to maintain steady straight direction of motion and the only impact is provided by the ice friction force, which is numerically equal to 4v2 Newtons, where the velocity v of the car is measured in m/sec. (a) Using Newton's Second Law F = ma, set up a mathematical model for the position x(t) and velocity v(t) of the car as functions of time t. Start by drawing a diagram and choosing a consistent system of units based on kg, m, sec (1 ton = 1000 kg, 1 m/sec 3.6 km/h, 1 N = 1 kg m/sec). Introduce and label the variables, show the units and write down the differential equations and the intial conditions. = (b) Use the model in part a to calculate v(t) and x(t). Fully show the process of solving the initial value problems. (c) Based on your work so far, how long will it take to pass the iced over stretch of the road and how fast will the car be moving as it exits the ice? Do not use a calculator! Leave your answers in an "algebraic" form, which may involve exponential functions and logarithms, simplify as appropriate, and clearly specify the units (e.g. sec, m/sec or km/h).