In a test report of the newspaper "auto motor sport" you could find the engine characteristics of the Audi R8, as shown on the left.
Using the coordinates for maximum torque and maximum power, determine an integer function for each of the following … … Have as small a grade as possible, … Describe the course of both graphs approximately and … In the extreme values exactly match the given values.
Note also the mathematical / physical relationship between M, n and P.
A cyclist rides on a mountainous track. It stops on a hilltop. During the first two minutes after this rest, its speed behaves according to the function:
V(t) = 0.0001 t3 – 0.02 t2 + 1.05t
1. How many seconds after the end of the break does the speedometer show the maximum speed?
2. What is the maximum speed reached by the cyclist in the considered period of time.
3. When and how large was the maximum acceleration?
4. When and how long was the maximum delay?
5. How many meters could the cyclist enjoy the acceleration?
6. How long does the cyclist need for the 1. Kilometers after the pause?
Solution: Task about path-time-speed-acceleration
A vehicle is briefly accelerated strongly from a standstill on a race track and then decelerated again to a standstill. In the first 20s. Of the trip, the speed behaves according to the function:
V(t) = 0.007*(t – 20)2 * t2
The SI units apply here, which are not written for clarity.
A: What is the maximum speed that the vehicle reaches?? B: How great is the highest acceleration? C: What track does the vehicle lay in these 20s. Back? C1: with TR C2: without TR D: Draw the 3 graphs s(t), v(t) and a(t).
Solution: Acceleration – Deceleration
On a section of a roller coaster the car moves as the function graph shows.
A: What is the highest acceleration? B: What is the maximum deceleration? C: What distance does the vehicle cover in these 6s? C1: with TR C2: without TR D: Draw the 3 graphs s(t), v(t) and a(t).
Standard solution: Roller coaster
Solution with a twist: Roller coaster
Solution with sine function: roller coaster
Task set, similar to a final exam
1. (10min) Exactly one injection into the intake manifold per cycle, with an injection time of 20ms. This corresponds to a duty cycle of 23. Determine the crankshaft speed. Note on the duty cycle: ( TV = ti / T ) —————————————————— Solution: nKW= 1380/min
2. For an older 6-cyl.-Gasoline engine with one ignition coil. Distributor the KW speed is 3280/min. Gasoline engine with one ignition coil. Distributor the KW speed is 3280/min. The closing angle under this condition is 56. Determine the closing time. —————————————————– Solution: nKW= 3,42ms
3. (10min) The engine from task. 2) has a speed range of (950… 6500) /min. The bore is 100mm and the stroke is 90mm. Estimate the maximum piston speed. (Make a note of your considerations and calculations.) ——————————————————— Solution: vKmax= ca. 32m/s calculated is the average piston speed at max. Engine speed However, the piston does not move uniformly but approximately sinusoidally, considering only the velocity magnitude. Consider only the upper sine half-wave and estimate the ratio of maximum to mean value to be about 1.7. So the maximum speed is about 1,6 times the average speed.
4. (40min) In a reciprocating machine, the piston force depends on the piston travel according to the following function:
F(s) = 3*10-6*( s2-380)2 *s2
The piston movement is bounded between the two outer zeros of this function. What is the maximum piston force? What is the maximum change in piston force per piston stroke? Determine the work done by the piston on its path. Draw the graph F(s) in the given limits. ————————————————————– Note on calculating the piston work: the work of the piston is calculated from the determined integral F(s)ds solutions: Fmax= 24.4 (units of force) F'max = 4.64 (units of force/displacement) W = 498 (units of work) Curve discussion table: … Draw graphs yourself : do it yourself
A tool slide is periodically moved back and forth on a guide. During a period, the speed of the sled takes place according to the function:
V(t) = 0.05*t*(t4 -12t2 +20)
(all units according to SI) This velocity function is valid between its two outer zeros.
Searched for: 1. Extreme values of velocity 2. Extreme values of acceleration 3. Total distance traveled during one hour of uninterrupted operation
Solutions (for your control): vE1 = 2.01 m/s at tE1= -2.57s vE2 = -0.51 m/s at tE1= -0.78s vE3 = 0.51 m/s at tE1= 0.78s vE4 = -2.01 m/s at tE1= 2.57s aE5 = -2.24 m/s^2 at tE5 =-1, 90s aE6 = 1.00 m/s^2 at tE6 = 0.00s aE7 = 2.24 m/s^2 at tE7 = 1.90s s = 5.20 m total travel during one period sges= 3603 m total travel in one hour
Given is the function
F(x) = 2×3 – 8×3 + 12
The two function terms of the semicircles are sought, which together give the full circle that touches the graph of f(x) at its inflection point from the left and has a diameter of 8 (units).
This one is pretty tough!! Without a sketch the task will probably remain incomprehensible for most students.
So : – Sketch a coordinate system – Plot the graph of a function 3. Degrees one. – Mark the point of inflection (WP). – Then sketch a circle so that it touches the turning point of f(x) from the left. If at the 1. If the first attempt does not work, at the latest at the 4th attempt, the second attempt will fail. Start up, you should have this sketch reasonably neat on paper. – Mark the center of the circle (M). – Sketch the horizontal diameter line of the circle. – denote the upper semicircle by fo(x). – denote the lower semicircle by fu(x). – Now draw the straight line that goes through M and WP. Now you already have a (hopefully) clean and readable plan sketch, on which you can study the relations vividly.
How to proceed now? Partial solutions for your control 1. Determine the coordinates of the turning point. WP (4/3 ; 68/27) 2. Since the circle should touch the function graph at the point of inflection, it must have the same slope there as f(x). F'(4/3) = -32/3 3. The center of the circle must lie on the straight line that is perpendicular to the turning tangent. 4. Determine the slope of this straight line. M = 3/32 5. Determine the function equation g(x) of the straight line through the M and WP. (point u. Slope of the straight line are given) g(x) = 3/32x + 2,39… 6. Since the diameter of the circle should be 8, M must be 4 units of WP on g(x).
7. So now we have a problem of linear functions. This secondary calculation becomes here performed. 8. The point M (-2.6492; 2.1416) lies to the left of point W on the straight line g and has a distance from W of 4.
9. The general equation of a circle with radius a and center (xM; yM) is: 10. (x – xm)2 + (y – ym)2 = a2 11. This equation must be rearranged to y. And 2 functions have to be formed, one of them with the positive root and the other one with the negative root. 12. This way you have a function for the upper semicircle respectively : yo(x) = root (16 – (x + 2.6492)2 ) + 2.1416 13. And a function for the lower semicircle : yu(x) = – root (16 – (x + 2.6492)2 ) + 2.1416
A motor vehicle starts at t=0 (s) and its speed follows the function
V(t) = 0.00003 * t2 * (t-30)2 * (t+10)
Here applies: [t]=s, [v]= m/s
Determine: 1. Time t1, where the vehicle again has the speed 0m/s. 2. Time t2 and value of maximum acceleration. 3. Time t3 and value of maximum velocity. 4. Time t4. Value of the maximum deceleration. Time t4 and value of the maximum deceleration. 5. Draw the graph v(t) in the interval [0; t1]. 6. Average speed of the motor vehicle in the interval [0; t1]. 7. The acceleration path and the acceleration time duration 8. The braking distance and the braking time duration
Note: The following applies: ds(t)/dt = v(t) and dv(t)/dt = a(t)
Solutions (for your reference):
v(t) = 0.00003 * (t5 – 50t4 + 300t3 + 9000t2) 1. Time t1, where the vehicle has the speed 0m/s again. (30s)
2. Time t2 and value of maximum acceleration. (9.04s; 3.66m/s^2)
3. Time t3 and value of maximum speed. 4. Time t4. Value of the maximum deceleration. Time t4 and value of maximum deceleration. (24.9s; -4.78m/s^2)
5. Draw the graph v(t) in the interval [0; t1]. (s. Diagram on the right)