APPLICATION OF INTEGRATION
1. A particle moves in a straight line. It passes through point O at t = 0 with velocity V = -4m/s. The
acceleration a m/s² of the particle at time t seconds after passing through O is given by a = 10t + 1
(a) Express the velocity V of the particle at time t seconds in terms of t. (3 marks)
b) Find V when t = 3 (1mark)
c) Determine the value of t when the particle is momentarily at rest (3marks)
d) Calculate the distance covered by the particle between t = 2 and t = 4 (3marks)
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2. The velocity of a moving particle in m/s in a straight line after 1 seconds is given by v = 4 + 8t - t 2
Calculate
a) The acceleration of the particle after 3 seconds. (2 marks)
b) The distance covered by the particle between t = 2 seconds and t = 6 seconds. (4 marks)
c) The time when the particle is momentarily at rest. (4 marks)
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3. The acceleration of a body moving along a straight line is (6-2t) m/s² and its velocity is vm/s after t
seconds.
a) i) If the initial velocity of the body is 6m/s, express the velocity V in terms of t. (3 marks)
ii) Find the velocity of the body after 5 seconds. (2 marks)
b) Calculate
i) the time taken to attain maximum velocity. (2 marks)
ii) the distance covered by the body to attain the maximum velocity. (3 marks)
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4. A particle starts from rest and its acceleration after t seconds is a = (35-t) m/s2. Calculate the
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distance this particle has covered at the instant when it reaches maximum velocity. (3 marks)
5. The velocity of a body moving on a straight line is given by by v=4t-t2
(a) If the initial displacement (s) in
(i) Meters of the body are 5m. Find displacement of the body after 3s. (3 marks)
(ii) Find the instances the body is momentarily at rest. (2 marks)
(b) Calculate the time taken for the body to attain maximum velocity. (2 marks)
(c) Calculate the displacement of the body at maximum height. (3 marks)
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6. (a) After t seconds, a particle moving along a straight line has a velocity of Vm/s and an acceleration
the particles initial velocity is 2m/s.
(i) Express V in terms of t. (3 marks)
(ii) Determine the velocity of the particle at the beginning of the third second. (2 marks)
(b) Find the time taken by the particle to attain maximum velocity and the distance it covered to
attain the maximum velocity. (5 marks)
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7. A particle moving along a line passes a point O at a velocity of 15m/s and its acceleration t seconds
later is given by a = (2t-8) m/s²
a) Find the expression of the velocity after passing the point O. (3 marks)
b) Find the time when the particle is at rest. (3 marks)
c) Find the distance between the points when the particle is at rest. (4 marks)
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8. A particle starts from rest and moves with an acceleration, a, given by a = (10-t) m/s². Given that
velocity, Vm/s in 2m/s, when time, t seconds is 1 sec.
a) Express in terms of t;
i) Its velocity after I seconds. (3 marks)
ii) Its displacement after t seconds. (2 marks)
b) Calculate its velocity when t = 3 seconds (2 marks)
c) Calculate the maximum velocity attained. (3 marks)
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9. A particle P moves in a straight line such that t seconds after passing a fixed point Q. it's velocity is
given by the equation 2t3-10t+ 12 find:
a) The values of t when p is instantaneously at rest. (2 marks)
b) An expression for the distance moved by P after t seconds. (2 marks)
c) The total distance traveled by P in the first 3 seconds after passing point O. (3 marks)
d) The distance of P from O when acceleration is zero. (3 marks)
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dy
10. a) The gradient function of a curve is given by = 2x²-5
dx
Find the equation of the curve, given that y = 3 and x = 2 (5 marks)
b) The velocity, Vm/s of a moving particle after t seconds is given by V = 2t3+ t2-1. Find the exact
distance covered by the particle in the interval 1≤t≤3 (5 marks)
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11. A particle moving with acceleration a = (10 - t) m/s2. When t=1 velocity V = 2 m/s and when t = 0
displacement S = 0M
a) Express displacement and velocity in terms of t. (3 marks)
b) Calculate the velocity when t = 35 (2 marks)
c) What is the displacement when t = 5 (3 marks)
d) Calculate maximum velocity. (2 marks)
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12. The acceleration of a particle in ms-2| given by the expressions 3t - 4
Find:-
(i) an expression for velocity Vms¹ (1 mark)
(ii) an expression for distance 5 metres from a fixed point O. Given that S=0 when V = 3 and t=0
(2 marks)
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13. A particle moves in a straight line. It passes through point O at t=0 with a velocity v = 5 m/s.
The acceleration a m/s² of the particle at time t seconds after passing through O is given by a = 6t+4
(a) Express the velocity v of the particle at time t seconds in terms of t. (3 marks)
(b) Calculate the velocity of the particle when t = 4. (2 marks)
(c) (i) Express the displacement s by the particle after t seconds in terms of t (2 marks)
(ii) Calculate the distance covered by the particle between t = 1 and t=4. (3 marks)
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14. A particle moves in a straight line such that its velocity v m/s is given by v =32+4t-t2 after t seconds.
Calculate
(a) Initial velocity (1mark)
(b) the acceleration when it comes to rest. (4marks)
(c) The distance traveled in the seventh second. (3marks)
(d) the total distance traveled. (2marks)
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15. The acceleration of a moving particle is given as a=6-2tm/s2. The velocity at the starting point is
8m/s Determine
(a) Its velocity in terms of t (2marks)
(b) The distance covered after 3 seconds (2marks)
(c) The distance covered during the 3rd second. (4marks)
(d) Maximum velocity of the particle (3marks)
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16. A particle P moves in a straight line so that its velocity, V m/s at time t≥0 seconds t is given by
V=28+t-2t2. Find
(a) The time when p is momentarily at rest. (3marks)
(b) The speed of P at the instant when the acceleration of the particles is zero. (4marks)
(c) Given that P passes through the point O of the line when t = 0, find the distance of P from O when P is
momentarily at rest. (3marks)
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17. The velocity of a particle moving in a straight line after t seconds is given by v= 4+8t-t 2 m/s.
Calculate
a) The acceleration of the particle after 3 seconds. (2marks)
b) The distance covered by the particle between 1-2 sec and t= 6 seconds. (4marks)
c) The time when the particle is momentarily at rest. (4marks)
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18. A particle starts from rest at a point A and moves along a straight line coming to rest at another point
B. During the motion, its velocity (vm/s) after time (t sec) is given by v = 9t2-2t3. Calculate:
a) the time taken for the particle to reach B. (2marks)
b) the distance traveled during the first two seconds. (3marks)
c) the time taken for the particle to attain its maximum velocity. (2marks)
d) the maximum velocity attained (3marks)
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19. The velocity of a particle is given as V = 12t-2t2.
a) determine the distance of the particle in terms of t if the distance is 6 metres. When t = 1 sec.
(2marks)
b) determine the distance moved by the particle during the third second. (2marks)
c) calculate the maximum distance moved by the particle. (2marks)
d) determine the acceleration after 2 seconds. (2marks)
e) when is the velocity maximum?
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