Add maths: Kinematics

Q1.A particle moves in a straight line and its displacement ,x metres, from a fixed point O, t seconds after passing O, is given by x= 12t - t^3.

(a)calculate the velocity of the particle when it is next at 0. [Ans: -24m/s]
(b)the distance travelled by the particle during the first 3 seconds. [ 23m]

Q2. A particle P travels in a straight line so that its displacement, x metres, from a fixed point O is given by x = 3t^2 - 4t^3 + 60, where t is the time in seconds measured from the start of the motion. Calculate:

the average speed of P over the first 2 seconds.[Ans: 10 1/4 m]



[ Formula to apply; velocity= ds/dt and acceleration = dv/dt]

Please help me on this, i'm so stressed out with this questions.Worse of all it kept repeating in the papers my teacher gave us,caused me to blank all these questions.....TT^TT so please help me on this

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Originally Posted by yumiryuiko

Q1.A particle moves in a straight line and its displacement ,x metres, from a fixed point O, t seconds after passing O, is given by x= 12t - t^3.

(a)calculate the velocity of the particle when it is next at 0. [Ans: -24m/s]
(b)the distance travelled by the particle during the first 3 seconds. [ 23m]

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To the first question:

The particle moves in such a way that first x increases, then the velocity inverts its sign and x decreases and goes back past 0. So, at t=0, the particle is at the origin, but after a time t₁:

0 = 12t₁ - t₁³
Solving:
t₁ = 2Sqrt(3)
The velocity is:
v = dx/dt = 12 - 3t²
and its value at that time is:
v(t₁) = 12- 3t₁² = 12 - 3(2Sqrt(3))² = -24

The distance travelled during the first 3 seconds is the sum of the distance traveled from t=0 until the velocity inverts its sign plus the distance traveled from that time until t=3. So, first you determine when the velocity inverts its sign, i.e. when it is 0:

0 = 12 - 2t² -> t = 2

Distance traveled from t=0 to t=2 (away from the origin):
x(+) = x(2) - x(0) = 16

Distance traveled from t=2 to t=3 (toward the origin):
x(-) = ABS[x(3) - x(2)] = ABS(9 - 16) = ABS(-7) = 7

Total = x(+) + x(-) = 16 + 7 = 23