IB · Mathematics: Analysis & Approaches SL & HL · Calculus

Kinematics: IB Mathematics: Analysis & Approaches questions with answers

Free Kinematics questions for IB Mathematics: Analysis & Approaches (SL & HL), part of Calculus. Each question has the answer and the marking points an examiner would look for.

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Kinematics practice questions

Question 1Structured · 6 marksA* challenge

The displacement s metres of a particle from a fixed point O, t seconds after it starts moving, is s(t) = 4te−0.5t − 1, t ≥ 0.

(a)

Find the times when the particle is at O.

[3]
(b)

Find the maximum distance of the particle from O, and the time at which this occurs.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)t = 0.289, t = 6.52
s(t) = 0 (M1)
t = 0.289 A1
t = 6.52 A1
3
(b)1.94 m at t = 2.00
s′(t) = 0 or graph (M1)
t = 2.00 A1
1.94 m A1
3
Question 2Structured · 7 marks

A particle moves in a straight line. Its velocity v m s−1 after t seconds is given by v(t) = (t − 2)(t − 4)e−0.5t, for 0 ≤ t ≤ 6.

(a)

Find the times when the particle is at rest.

[2]
(b)

Find the total distance travelled by the particle for 0 ≤ t ≤ 6.

[3]
(c)

Find the displacement of the particle at t = 6, relative to its starting position.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)t = 2, t = 4
v = 0 (M1)
t = 2, t = 4 A1
2
(b)5.82 m
∫06|v(t)|dt (M1)(A1)
5.82 m A1
3
(c)5.21 m
∫06v(t)dt (M1)
5.21 m A1
2
Question 3Structured · 13 marks

A particle moves in a straight line. Its velocity v m s−1 at time t seconds is given by v(t) = 4e−t/4sin t, for 0 ≤ t ≤ 2π.

(a)

Find the times when the particle is at rest.

[2]
(b)

Find the maximum speed of the particle, and the time at which it occurs.

[3]
(c)

Find the acceleration of the particle when t = 2.

[2]
(d)

Find the total distance travelled by the particle.

[3]
(e)

Find the displacement of the particle at t = 2π and explain why it differs from the distance travelled.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)t = 0, π, 2π
v = 0 ⇒ sin t = 0 (M1)
t = 0, π, 2π A1
2
(b)t = 1.33 s, 2.79 m s−1
use of graph or solving v′(t) = 0 (M1)
t = 1.33 A1
2.79 m s−1 A1
3
(c)−1.56 m s−2
v′(2) (M1)
−1.56 m s−2 A1
2
(d)7.98 m
∫02π|v(t)|dt (M1)(A1)
7.98 m A1
3
(e)2.98 m
∫02πv(t)dt (M1)
2.98 m A1
the particle changes direction at t = π, so displacement < distance R1
3
Question 4Structured · 6 marksA* challenge

The displacement s metres of a particle from a fixed point O, t seconds after it starts moving, is s(t) = 4te−0.3t − 1, t ≥ 0.

(a)

Find the times when the particle is at O.

[3]
(b)

Find the maximum distance of the particle from O, and the time at which this occurs.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)t = 0.271, t = 13.2
s(t) = 0 (M1)
t = 0.271 A1
t = 13.2 A1
3
(b)3.91 m at t = 3.33
s′(t) = 0 or graph (M1)
t = 3.33 A1
3.91 m A1
3
Question 5Structured · 7 marks

A particle moves in a straight line. Its velocity v m s−1 after t seconds is given by v(t) = (t − 1)(t − 3)e−0.3t, for 0 ≤ t ≤ 4.

(a)

Find the times when the particle is at rest.

[2]
(b)

Find the total distance travelled by the particle for 0 ≤ t ≤ 4.

[3]
(c)

Find the displacement of the particle at t = 4, relative to its starting position.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)t = 1, t = 3
v = 0 (M1)
t = 1, t = 3 A1
2
(b)2.40 m
∫04|v(t)|dt (M1)(A1)
2.40 m A1
3
(c)0.921 m
∫04v(t)dt (M1)
0.921 m A1
2
Question 6Structured · 7 marks

A particle moves in a straight line with velocity v(t) = 2t2 − 6t + 3.50 m s−1, for 0 ≤ t ≤ 6.

(a)

Find the times when the particle is at rest.

[2]
(b)

Find the acceleration when t = 2.

[2]
(c)

Find the total distance travelled in the first 6 seconds.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)t = 0.793, 2.21
v = 0 (M1)
t = 0.793, 2.21 A1
2
(b)2.00 m s−2
differentiating (M1)
2.00 A1
2
(c)58.9
∫|v| dt (M1)
correct limits (A1)
answer A1
3

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in