Maths assignment Problems
KnowledgeCats
3. (a) A projectile fired at an angle α has its horizontal (x) and vertical (y) distance as a function of time, t, given as:
x = utcos(α)
y = utsin(α) − gt22
where g is the acceleration due to gravity, and u is the initial velocity of the projectile.
At what time does the projectile reaches the highest point in its trajectory? Express your answer in terms of u, g, and α
(b) Similarly, determine the maximum height.
(c) At what time after launch does the projectile hit the ground?
(d) Evaluate dy and d 2 y
dx dx 2
[3 marks] [3 marks] [3 marks]
[6 marks]
(e) Use that u = 100 m/s and g = 10 m/s2. Make a sketch of the trajectory of the projectile for a launch at an angle with the horizon of α= 30° and in the
same sketch, add the trajectory for a launch angle α= 70°
(f) Show that the horizontal distance travelled by the projectile can be written as:
x = 2gu2sin α cos α
For which value of α does the projectile travels furthest?
4. | (a) | Let z = 1 + j | and 1 − j be two complex numbers. Determine the following: |
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(b) Express the complex number −4 in modulus-argument form. Hence find all solutions z to the equation
z4+4=0and mark them on an Argand diagram.
(c)
[5 marks]
[5 marks]
[9 marks]
[9 marks]
Page 3 of 5
5 (a)
(b)
(c)
(d)
Consider a simple harmonic oscillator whose motion is illustrated in the diagram above. We can describe the motion in complex form as
P (t )= Re jωt.
Determine P (t)= Re jωt in rectangular form for R = 2 and
(i) ωt= | π | (ii) ωt= | π | (iii) ωt=π |
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Solve the following linear system of equations: |
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10x | + | y | − 5z | = | 18 | |
−20x + 3y + | 20z | = 14 | ||||
5x | + | 3 y | + | 5z | = | 9 |
Determine AB if possible for the following matrices:
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(a) | B = | (1 | 2 |
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A = | , |
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(c) | A =(1 |
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2) , B = | 4 |
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[7 marks]
10 marks
8 marks
2 marks
5 marks
Page 4 of 5
(a) Evaluate the following:
∫24 x3dx
∫42 x3dx
6 marks
(b) Let f ( x)=∫03 ex2 dx Determine the derivative f ′( x) .
4 marks
(i) Show that
dxd sin (x)ln (x)= cos ( x) ln ( x)+sinx(x)
4 marks
Find
∫ | cos ( x) ln ( x)+ | sin( x)dx |
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3 marks
- 10 years ago
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- maths_assignment_problems_solution.docx
- maths_assignment_problems_solution_6a_abd_b.docx