Topic 1

To understand how velocity and acceleration are applied to one-dimensional motion, consider a spacecraft equipped with two engines that are mounted perpendicular to each other. These engines produce the only forces that the craft experiences. In the following figure, only the engine oriented along the x direction is firing, and the vehicle accelerates along this direction. It is assumed that the velocity in the y direction is zero, and it remains zero, since the y engine is turned off. Its acceleration is given by a(t) = 1 e^(-t). Find its velocity along this direction, from t = 0 s to t = 3 s by calculating the following integral

?(1-e^(-t) ) dt from t = 0 to t = 3

(i) Evaluate the above integral analytically

(ii) Evaluate the above integral using the single application of the trapezoidal rule

(iii) Evaluate the above integral using the multiple application of the trapezoidal rule, with n = 2, 4 and 6

(iv) Evaluate the above integral using the single application of Simpsons 1/3 rule

(v) Evaluate the above integral using the multiple-application of Simpsons 1/3 rule, with n = 4

(vi) Evaluate the above integral using the multiple-application of Simpsons rule with n = 5 (Simpsons 3/8 to be used in conjunction with Simpsons 1/3 rule)

For each of the numerical estimates (ii) through (vi), determine the percent relative error based on (i). By calculating the local error O(h^k) for both Simpsons 1/3 and 3/8 rules, describe which of these two methods is more accurate. Point out the particular cases, where we prefer Simpsons 3/8 usage?

You are at the wheel of a 1200 kg car travelling east through an intersection when a 3000 kg truck travelling north through the intersection crashes into your car, as shown in the following figure. Your car and the truck remain stuck together after impact. The driver of the truck claims you were at fault because you were speeding. You look for evidence to disprove this claim. First, there are no skid marks, indicating that neither you nor the truck driver saw the accident coming and braked hard; second, the posted speed limit for the road on which you were driving is 80 km/h; third, the speedometer of the truck was smashed on impact, leaving the needle stuck at 50 km/h; and fourth, the wreck initially skidded from the impact zone at an angle of 59 degrees north of east. Does this evidence support or undermine the claim that you were speeding?

4.2 You have a summer job with Goodyear as part of an automobile tyre testing team. You are testing a new model of racing tyres to see whether or not the coefficient of static friction between the tyres and dry concrete pavement is 0.90 as claimed by the manufacturer. In a skidpan test, a race car is able to travel at constant speed in a circle of radius 45.7 m in 15.2 s without skidding. Assume air drag and rolling friction are negligible and assume that the road surface is horizontal. In a skidpan test a car travels in a circle on a flat, horizontal surface (a skidpan) at the maximum possible speed v without skidding.

What was its speed v?

What was the acceleration?

What was the minimum value for the coefficient of static friction between the tyres and the road?