| Given a displacement equation |
1. A particle moves along a straight line so that its position
s(t) cm at time t minutes is given by
s(t) = t3 - 12t2 + 36t
(i) Find an expression for the velocity v(t) and acceleration a(t) of the particle in terms of t.
(ii) When is the particle at rest?
(iii) Complete the following table with the times t = 0, 2, 4, 6 and 8 and calculate the missing values for s(t), v(t) and a(t).
| t |
s(t) |
v(t) |
a(t) |
| 0 |
0 |
36 |
-24 |
| 2 |
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| 4 |
16 |
-12 |
0 |
| 6 |
0 |
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| 8 |
32 |
36 |
24 |
(iv) Sketch all 3 graphs and describe the motion of the particle in the first 8 seconds.
Answer.(i) v =3t2-24t + 36.
(ii) At t = 2 and 6 seconds. |
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2. A particle is moving in a straight line such that its displacement x metres from a fixed point O is given by
x(t) = 3t2 - t3 for t ≥ 0.
(i) Find the initial velocity and acceleration particle.
(ii) Find the stationary points on the displacement-time graph and state their nature.
(iii) Determine the time at which the acceleration of the particle is zero.
(iv) Graph the function x(t), t ≥ 0.
(v) Determine the total distance travelled by the particle in the first 4 seconds.
(vi) Describe the motion of the particle.
Answer.(i) v = 0 and a = 6.
(ii) At t = 0, x = 0 (min)
at t = 2, x = 4 (max)
(iii) acc = 0 when t = 1 sec.
(iv) x = 24 metres.
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3. A particle moves such that its position, x metres, from a fixed point O is given by the function x = t3 - 7.5t2 + 18t + 2
where t is measured in seconds.
(i) Find the particle's initial position and velocity.
(ii) When is the particle at rest?
(iii) What is the acceleration of the particle when it is first at rest?
(iv) Find the distance travelled by the particle in the first three seconds.
Answer.(i) Particle at +2 m with velocity of 18 m/sec.
(ii) At rest when t = 2 and t = 3.
(iii) Distance = 14.5 m.
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4. The displacement of a particle from the origin, x metres, after t seconds is given by the equation x = t(t - 6)2
(i) Find the initial position of the body.
(ii) When is the body again at this position?
(iii) Obtain the velocity and accelerations functions for the motion.
(iv) At what times is the body momentarily at rest?
(v) In which direction does the body move between these times?
(vi) What is the initial acceleration for the motion? Is it tending to slow down or speed up the initial movement?
(vii) At what instant is there no acceleration acting on the moving body? What is the velocity at that instant? Show that this is the greatest speed in a negative direction attained by the moving body.
(viii) Draw graphs to show each of the displacement, velocity and acceleration functions.
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5. The Menara Tower in Kuala Lumpur is the fourth tallest telecommunications tower in the world. It contains a lift to take tourists directly from the base to the observation deck. At time t seconds after it leaves the base, the height h metres of the lift above the base is given by
(i) Find an expression for the velocity of the lift after t seconds.
(ii) The lift is initially at rest and it comes to rest again when it reaches the observation deck. How many seconds does the lift take to reach the observation deck?
(iii) Find the height of the observation deck above the base.
(iv) Find when the lift is travelling fastest and its speed at the time.
(v) Sketch the graph of height against time for 0 ≤ t ≤ 60.
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| Log function |
6. A particle moves in a straight line so that its displacement, x metres, at time t seconds is given by
x = 6t + 3 loge (3t2 + 1).
(i) Find the initial displacement.
(ii) Prove that the particle is never at rest.
(iii) Hence find the limiting velocity.
Answer.(i) x = 0.
(ii) v = 6+3t/(3t2+1)> 0 all t
(iii) lim vel when t approaches infinity - so 6 m/sec. |
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7. A particle is moving according to the function
x = 9t3 - ln t (for t > 0).
The displacement (x) is measured in metres and time (t) in seconds.
(i) Find an expression for the velocity of the particle.
(ii) Hence find when the particle comes to rest.
(iii) Explain why the acceleration remains positive for all values of t > 0.
Answer.(i) v = 27t2 - 1/t
(ii) At t = 1/3 sec.
(iii) Acc = 54t +1/t2 > 0
for all t > 0.
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8. |
| Exponential eqn. |
9. A particle moves along the x-axis so that its displacement x metres after t seconds is given by  .
(i) Show that the velocity is given by
 .
(ii) When does the particle first come to rest?
(iii) Show that the acceleration is zero when
 .
(iv) Find the particle's maximum speed correct to 2 decimal places.
(v) What happens to the particle eventually?
Answers(ii) 1st at rest at t = 0.6 secs.
(iv) Maximum speed is 6.3 m/sec.
(v) The particle approaches the origin from the right at decreasing speed. |
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10. 
A particle is moving in a straight line motion. The particle starts from the origin and, after t seconds, it has displacement of x metres from O. The equation for the displacement is
The particle has velocity v m/sec. which is given by
(i) What is the initial velocity of the particle?
(ii) When and where will the particle be at rest?
(iii) Find the time when the particle has zero acceleration.
(iv) What is the time interval when the particle is increasing its speed?
Answers(i) Init vel. = 4 m/sec.
(ii) At rest at 2 secs at 8e-1 m.
(iii) Zero accel at 4 secs.
(iv) 2 < t < 4
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11. A particle moves in a straight line so that its distance x metres from a fixed point O is given by x = 2t + e-2t where t is measured in seconds.
(i) What is the velocity of the particle when t = 0.5 seconds?
(ii) Show that the particle is initially at rest.
(iii) Find the limiting velocity of the particle as t increases.
(iv) Using v as the velocity and a as the acceleration, show that
a = 4 - 2y.
Answer.(i) v = 2 - 2/e.
iii) limiting vel = 2 m/sec.
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| Trig function. |
12. A particle is moving along the x-axis. Its position at time t is given by x = t + sin t.
(i) At what times during the period 0 < t < 3π is the particle stationary?
(ii) At what times during the period 0 < t < 3π is the acceleration equal to zero?
(iii) Sketch the graph x = t + sin t for 0 < t < 3π. Clearly label any stationary points and any points of inflexion.
Answers(i) Stationary at t = π sec.
(ii) Acceleration = 0 at t = π and at 2π.
(iii) Careful of the POI at t = π.
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13. A particle moves in a straight line from a fixed point O. Its displacement x cm at time t seconds is given by
x = 1 - 2cos 2t.
(i) Find the initial position of the particle.
(ii) Find the first time the particle is at the origin
(i.e. when x = 0).
(iii) Sketch the graph of x as a function of t for 0 ≤ t ≤ 2π.
(iv) Initially the particle is at rest. Find the next time the particle is at rest.
Answer.(i) Particle is at -1 cm.
(ii) time at the origin = π/6.
(iv) Next time is at π/2 secs.
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14. A particle moves along a horizontal line so that its displacement x cm to the right of the origin at any time t seconds is x = t sint.
(i) Find expressions for the velocity and acceleration of the particle.
(ii) Find the exact velocity of the particle at time t = π/4.
(iii) What effect does the acceleration have on the velocity of the particle at time t = π/4.
(iv) After the particle leaves the origin, is the particle ever at rest? Give reasons for your answer.
Answer.(i) vel = sint + t cost;
accel - 2 cost - t sint.
(ii) v = 1/√2 + π/(4√2)
(iii) Accel is speeding up the particle.
(iv) Draw the graphs of tan t = -t; so yes it does get to the origin again.
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| Given velocity equation. |
15. The velocity of a particle moving along the x-axis is given by
velocity = 27t - 15t2
where x is the displacement from the origin in metres and t is the time in seconds.
Initially the particle is at the origin.
(i) After t = 0, show that the particle comes to rest again at
t = 1.8 seconds.
(ii) What is the velocity of the particle when the acceleration is zero?
(iii) Sketch the graph of the acceleration function against time.
(iv) If the particle continues on the same path, when will it return to the origin?
Answer.(i) After t = 0 is at 27/15 = 1.8 secs.
(ii) Acc = 0 when t = 0.9 secs so v = 12.15 m/sec.
(iv) At the origin again after 2.7 secs.
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16. A train stops at two stations. Its velocity between those two stations is given by v = 0.5t (2 - t) with velocity in km/hour and time in minutes.
(i) What is the time taken between the two stations?
(ii) What is the maximum velocity between the two stations and when was it attained?
(iii) What is the distance between the two stations?
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17. The velocity v metres/sec of a particle t seconds after starting to move is given the equation  .
(i) Sketch the graph of the velocity function.
(ii) Find the distance travelled by the particle in the first 3 seconds.
(iii) Find the distance travelled by the particle in the next three seconds.
(iv) Find the displacement of the particle from its initial position in the first six seconds.
Answer.(ii) 4.5 m right.
(ii) 18 m to the left.
(iii) Distance = 18 m. |
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18. The velocity v m/sec of an object at time t seconds is given by v = 3t2 - 14t + 8. The object is initially 30 m to the right of the origin.
(i) Find the initial acceleration of the object.
(ii) Find when the object is at rest.
(iii) Find the minimum distance between the origin and the object during its motion.
Answer.(i) Initial accel is 14 m/sec2.
(ii) At rest after 2/3 or 4 seconds.
(iii) Min distance = 14 m.
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19. The velocity of a particle moving is a straight line is given by
v = (t - 2)2 + 5 m/sec.
(i) Find the minimum speed of the particle.
(ii) Find the initial acceleration of the particle.
(iii) Find the distance travelled by the particle in the first 3 seconds.
(iv) Initially the particle is at 9 m to the left of the origin. Find the position of the particle at t = 5 secs.
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20. The velocity of a particle moving along the x-axis is given by  where x is the displacement from the origin in metres and t is the time in seconds. Initially the particle is at the origin.
The diagram shows the graph of the velocity.
(i) Show that the particle comes to rest again at t = 1.8 seconds.
(ii) What is the velocity of the particle when the acceleration is zero?
(iii) Sketch the graph of the acceleration function.
(iv) If the particle continues on the same path, when will it return to the origin?
Answer.(ii) At t = 0.9 secs, v = 12.15 m/sec.
(iv) Return to origin at 2.7 secs. |
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21. A particle moves in a straight line. Its velocity v m/sec at time t seconds is given by  .
The particle is at rest 4 m to the right of the origin after  seconds.
(i) Find the displacement after  seconds.
(ii) Find the total distance travelled in the first seconds.
(iii) What is the maximum acceleration and when will the particle first experience this acceleration?
Answer.(i) At 4.5 m to the right of the origin.
(ii) Distance is (4-√3)/4 m.
(iii) Max accel is 2 m/sec2
at t = π/12 secs.
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| Given acceleration equation. |
22. A particle with acceleration of a = 6t +4 is initially at rest at the origin.
Find the equation for the displacement of the particle.
Answer.x = t.3 + 2t2. |
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23. A particle moves in a straight line with acceleration given by
 .
Find the equation for the displacement given that when
t = 0, x = 0 and  |
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24. The acceleration of a particle is given by  where x is displacement in metres and t is time in seconds.
Initially the particle was 1 metre to the left of the origin and was moving with a velocity of 2 metres per second to the right.
(i) Determine the equations for both the velocity of the particle and the displacement of the particle.
(ii) Find the distance travelled by the particle in the first four seconds.
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25.
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26. A particle is moving with acceleration described by
 . It is initially at the origin and moving with a velocity of 2 m/sec.
(i) By determining velocity equation, show that the particle was stationary for the first time at t = 3π seconds.
(ii) How far did the particle move before it became stationary?
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| Two particles |
27. Two particles A and B start from the origin at the same time. They move along a straight line with their velocities, in m/sec, at any time t being described by the equations:
vA = t2 + 2 and vB = 8 - 2t.
(i) Show that the particles never move with the same acceleration.
(ii) Write an equation to describe the position of particle A at time t seconds.
(iii) After leaving the origin, determine when and where the two particles are together again.
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28. Two particles A and B move along the x axis - both starting when t = 0.
The displacement equations for the two particles are:
xA = t + 12 - t2 and xB = t2 - 4t
where the displacement x in both cases is from the origin.
(i) Find when and where particle A is stationary.
(ii) On the same diagram, sketch each particles displacement graph. Label the main features.
(iii) Show that the distance D between the two particles during 0 ≤ t ≤ 4 is given by
S = 5t + 12 - 2t2.
(iv) During the first 4 seconds, when are the particles furthest apart?
(v) Find the time when both particles have the same velocity.
(v) What conclusion can you state about the accelerations of each of the two particles. Support your conclusion with some good maths!!
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29. Two particles A and B start moving on the axis at the same time t = 0. The position of particle A at time t can be described by
and the position of particle B at time t is given by
(i) Find expressions for the velocities of each of the two particles.
(ii) Given that there are two occasions when the two particles have the same velocity, show that the distance travelled by particle A between the two occasions is
(iii) Show that the two particles never meet.
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30. A ball falls from rest in a fluid medium with its acceleration described by:
where x metres is the dustance below the origin at time t seconds.
(i) Find the velocity-time function for the motion of the ball and sketch it.
(ii) What is the limiting velocity of the ball?
(iii) How far does the ball travel in the first 3 seconds?
(iv) A second ball moves with the same acceleration function as the first ball but it is initially thrown upwards with speed 10 m/sec from a point 2 metres below the origin at the same instant that the first ball is dropped. Do the balls ever collide and, if they do collide, at what time?
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