IB Maths AI HL 5.13 Kinematics

[Maximum number: 11]

A sports stadium has a T-shirt cannon which is used to launch T-shirts into the crowd. The purpose of this question is to determine whether a person sitting in a particular seat will ever receive a T-shirt.
A T-shirt cannon is placed on the horizontal ground of a stadium playing area. A coordinate system is created such that the origin, O , is the point on the ground from where the T-shirts are launched. In this coordinate system, x and y represent the horizontal and vertical displacement from O, and are measured in metres.
Seat $\mathrm{A}_{1}$ is the nearest seat to the T-shirt cannon. The coordinates of the front of the foot space for seat $\mathrm{A}_{1}$ are (30, 2.1).

Figure for Question AHL 5.13 (HL)—Kinematics question 1 — IB Maths AI HL

Each seat behind seat $\mathrm{A}_{1}$ is 1.0 m further from O horizontally and 0.5 m higher than the seat in the row below it, as shown on the diagram.

Seat $\mathrm{A}_{1}$ is in row 1 . Let seat $\mathrm{A}_{n}$ be the seat directly behind $\mathrm{A}_{1}$ in row n.

(a)

Find an expression for the velocity, $\binom{\dot{x}}{\dot{y}}$ , at time t.

[ 3 ]

evidence of integration of the acceleration vector OR use of v=u+a t (M1)

\begin{aligned} & \binom{\dot{x}}{\dot{y}}=\binom{c_{1}}{-9.8 t+c_{2}} \\ & \left(\binom{\dot{x}}{\dot{y}}=\right)\binom{29.4 \cos \theta}{29.4 \sin \theta-9.8 t} \end{aligned}

Note: The first A1 is for $\dot{x}$ and the second is for $\dot{y}$ .

(b)

Hence show that when the T-shirt is launched vertically, the time for it to reach its maximum height is 3 seconds.

The displacement of the T-shirt, t seconds after it is launched, is given by the vector equation

\binom{x}{y}=\binom{29.4(\cos \theta) t}{29.4(\sin \theta) t-4.9 t^{2}}

[ 3 ]

$\theta=90^{\circ}$

29.4-9.8 t=0

Note: Award M1 for setting their $\dot{y}$ to zero (may still include $\theta$ ), A1 for correct equation, leading to given result.
If they substitute t=3 award at most (A1)MOAO.

\begin{aligned} & \text { maximum point when } t=\frac{29.4}{9.8} \\ & =3 \text { (seconds) } \end{aligned}

(c)

Using the given answer to part (b)(ii) or otherwise, find the maximum height reached by a T-shirt when it is launched vertically.

[ 2 ]

correct substitution OR use of correct graph maximum height is $29.4 \times 3-4.9 \times 3^{2}$

\text { = } 44.1 \text { (m) }

(d)

If there was no seating, and the T-shirt was launched at an angle $\theta$ , show that the value of x when it would hit the ground is given by the expression

x=176.4 \sin \theta \cos \theta

[ 3 ]

$\quad 29.4 \sin \theta t-4.9 t^{2}=0$ $t=6 \sin \theta \quad($ or t=0)
$x=29.4 \cos \theta \times 6 \sin \theta$ $=176.4 \cos \theta \sin \theta$

IB Maths AI HL 5.13 Kinematics Question Bank

Exam points

AHL 5.13 (HL)—Kinematics question 1

Question (a)

Question (b)

Question (c)

Question (d)