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Worksheet
Kinematics
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Name
Question 1
A particle moves along a straight line. Its displacement from the origin at time \(t\) seconds is \(s(t)=t^3-6t^2+9t+2\), \(t\geq0\).
(a)Find the velocity and acceleration functions.
(b)Find when the particle is stationary.
(c)Find the displacement when \(t=4\).
(d)Find the total distance travelled in the first 4 seconds.
(a)
Answer
\(v(t)=3t^2-12t+9\); \(a(t)=6t-12\)
(b)
\(v=0\)
\(3t^2-12t+9=3(t-1)(t-3)=0\)
Answer
\(t=1\) s and \(t=3\) s
(c)
Answer
\(s(4)=64-96+36+2=6\) m
(d)
Track direction changes
\(s(0)=2;\;s(1)=6;\;s(3)=2;\;s(4)=6\)
Distance
\(|6-2|+|2-6|+|6-2|=4+4+4\)
Answer
12 m
Question 2
A particle has velocity \(v(t)=4\sin(2t)\) m/s. At \(t=0\) the particle is at \(s=3\) m.
(a)Find the displacement function.
(b)Find the acceleration when \(t=\dfrac{\pi}{4}\).
(a)
Integrate
\(s=\displaystyle\int4\sin(2t)\,dt=-2\cos(2t)+C\); \(s(0)=-2+C=3\Rightarrow C=5\)
Answer
\(s(t)=5-2\cos(2t)\)
(b)
Differentiate \(v\)
\(a(t)=8\cos(2t);\;a\!\left( frac{\pi}{4} ight)=8\cos\!\left( frac{\pi}{2} ight)=0\)
Answer
\(0\) m/s²
Question 3
A ball is thrown upward from 2 m above the ground with initial velocity 15 m/s. Its height is \(h(t)=2+15t-4.9t^2\).
(a)Find the maximum height reached.
(b)Find when the ball hits the ground.
(a)
\(h'(t)=15-9.8t=0\Rightarrow t= frac{15}{9.8}\)
\(h=2+15\cdot frac{15}{9.8}-4.9\!\left( frac{15}{9.8} ight)^2pprox13.5\)
Answer
Approx \(13.5\) m
(b)
Solve \(h(t)=0\)
Using quadratic formula: \(t=\dfrac{-15\pm\sqrt{225+4 imes4.9 imes2}}{-9.8}pprox3.19\) s
Answer
Approx \(3.19\) s
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