Exam-Style Questions.

1.	IB Standard

The acceleration, \(a\) ms^-2 , of an object at time \(t\) seconds is given by

$$a=\frac1t+4sin3t, (t\ge1)$$

The object is at rest when \(t=1\).

Find the velocity of the object when \(t=7\).

Worked Solution

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2.	IB Standard

Very accurate equipment was used to measure the movement of a particle which moved in a straight line for 3 seconds. Its velocity, \(v\) ms^-1 , at time \(t\) seconds, is given by:

$$v=(t^2-5)^3$$

(a) Find the velocity of the particle when \(t=2\).

(b) Find the value of t for which the particle is at rest.

(c) Find the total distance the particle travels during the first three seconds.

(d) Show that the acceleration of the particle is given by \(a=6t(t^2-5)^2\)

(e) Find all possible values of t for which the velocity and acceleration are both positive or both negative.

Worked Solution

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3.	IB Analysis and Approaches

A particle moves in a straight line. During the first nine seconds the velocity, \(v\) ms^-1 of the particle at time \(t\) seconds is given by:

$$ v(t) = t \cos(t+5) $$

The following diagram shows the graph of v:

(a) Find the maximum value of \(v\).

(b) Find the acceleration of the particle when t = 6.

(c) Find the total distance travelled by the particle.

Worked Solution

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4.	IB Standard

Pob and Wie are travelling from Bangkok to Khon Kaen.

Pob travels at a velocity given by \(V_P=50-t^2\), where t is in seconds and the velocity is in ms^-1.

Wie's displacement from Bangkok in metres is given by \(S_W=2t^2+70\).

When \(t=0\), both vehicles are at the same point.

Find Pob's displacement from Bangkok when \(t=5\).

Worked Solution

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5.	IB Analysis and Approaches

A particle travels in a straight line such that its displacement, \(s\) metres, from a fixed point, \(A\), after \(t\) seconds is given by \(s(t) = t^2 - 5t \), for \( 0 \le t \le 10\), as shown in the following sketch.

The particle starts at \(A\) and passes through \(A\) again when \(t = q\).

(a) Find the value of \(q\).

The particle changes direction when \(t = r\).

(b) Find the value of \(r\).

(c) Find the displacement of the particle from \(A\) when \(t = r\).

(d) Find distance of the particle from \(A\) when \(t=10\).

(e) Find the total distance travelled by the particle in ten seconds.

A second particle travels along the same straight line such that its velocity is given by \(v(t)=12 - 3t\) for \(t \ge 0\).

When \(t = p\), the distance travelled by this second particle is one metre less than the total distance travelled by the first particle in ten seconds.

(f) Find the value of \(p\)

Worked Solution

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6.	IB Analysis and Approaches

A particle P moves along a straight line. The velocity of P is \(v\)ms^-l at time \(t\) seconds, where \(v(t) = 5 + 3t - 2t^2\) for \(0 \le t \le 3\). When \(t = 0\), P is at the origin O.

(a) Find the value of \(t\) when P reaches its maximum velocity.

(b) Find the distance of P from O at this time.

(c) Sketch a graph of \(v\) against \(t\), clearly showing any points of intersection with the axes.

(d) Find the total distance travelled by P.

Worked Solution

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7.	IB Standard

Make a sketch of a graph showing the velocity (in \(ms^{-1}\)) against time of a particle travelling for six seconds according to the equation:

$$v=e^{\sin t}-1$$

(a) Find the point at which the graph crosses the \(t\) axis.

(b) How far does the particle travel during these first six seconds?

Worked Solution

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8.	IB Standard

A particle P moves along a straight line. The velocity \(v\) in metres per second of P after \(t\) seconds is given by \(v(t) = 3\sin{t} - 8t^{\cos{t}}, 0 \le t \le 7\).

(a) Find the initial velocity of P.

(b) Find the maximum speed of P.

(c) Write down the number of times that the acceleration of P is 0 ms^-2.

(d) Find the acceleration of P at a time of 5 seconds.

(e) Find the total distance travelled by P.

Worked Solution

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