A particle starts from a point O and moves in a straight line. The velocity v m s-1 of the particle at time t s after leaving O is given by
\(v = k(3t^2 - 2t^3)\),
where k is a constant.
Find k and hence find the total distance travelled in the first two seconds of motion.
A cyclist starts from rest at a fixed point O and moves in a straight line, before coming to rest k seconds later. The acceleration of the cyclist at time t seconds after leaving O is a m/s2, where a = 2t - \frac{3}{5}t^2 for 0 < t \leq k.
A particle P moves in a straight line, starting from rest at a point O on the line. At time t s after leaving O the acceleration of P is k(16 - t^2) m s-2, where k is a positive constant, and the displacement from O is s m. The velocity of P is 8 m s-1 when t = 4.
A cyclist starts from rest at a point A and travels along a straight road AB, coming to rest at B. The displacement of the cyclist from A at time t s after the start is s m, where
\(s = 0.004(75t^2 - t^3)\).
(a) Show that the distance AB is 250 m.
(b) Find the maximum velocity of the cyclist.
A particle moving in a straight line starts from rest at a point A and comes instantaneously to rest at a point B. The acceleration of the particle at time t s after leaving A is a m s-2, where
\(a = 6t^{\frac{1}{2}} - 2t\).
A particle moves in a straight line. It starts from rest from a fixed point O on the line. Its velocity at time t s after leaving O is v m s−1, where v = t2 − 8t3/2 + 10t.
\((a) Find the displacement of the particle from O when t = 1.\)
(b) Show that the minimum velocity of the particle is −125 m s−1.
A particle P moves in a straight line. It starts at a point O on the line and at time t s after leaving O it has velocity v m s-1, where v = 4t^2 - 20t + 21.
(a) Find the values of t for which P is at instantaneous rest.
(b) Find the initial acceleration of P.
(c) Find the minimum velocity of P.
(d) Find the distance travelled by P during the time when its velocity is negative.
A particle P moves in a straight line, starting from a point O with velocity 1.72 m s-1. The acceleration a m s-2 of the particle, t s after leaving O, is given by a = 0.1t3/2.
(a) Find the value of t when the velocity of P is 3 m s-1.
\((b) Find the displacement of P from O when t = 2, giving your answer correct to 2 decimal places.\)
A particle P moves in a straight line. It starts from rest at a point O on the line and at time t s after leaving O it has acceleration a m s-2, where a = 6t - 18.
Find the distance P moves before it comes to instantaneous rest.
A particle travels in a straight line PQ. The velocity of the particle t s after leaving P is v m s-1, where
\(v = 4.5 + 4t - 0.5t^2\).
A particle moves in a straight line starting from a point O before coming to instantaneous rest at a point X. At time t s after leaving O, the velocity v ms-1 of the particle is given by
\(v = 7.2t^2 \quad 0 \leq t \leq 2,\)
\(v = 30.6 - 0.9t \quad 2 \leq t \leq 8,\)
\(v = \frac{1600}{t^2} + kt \quad 8 \leq t,\)
where k is a constant. It is given that there is no instantaneous change in velocity at \(t = 8\).
Find the distance OX.
A particle moves in a straight line AB. The velocity \(v \text{ m s}^{-1}\) of the particle \(t\) s after leaving A is given by \(v = k(t^2 - 10t + 21)\), where \(k\) is a constant. The displacement of the particle from A, in the direction towards B, is 2.85 m when \(t = 3\) and is 2.4 m when \(t = 6\).
A particle moves in a straight line through the point O. The displacement of the particle from O at time t s is s m, where
\(s = t^2 - 3t + 2\) for \(0 \leq t \leq 6\),
\(s = \frac{24}{t} - \frac{t^2}{4} + 25\) for \(t \geq 6\).
Particle P travels in a straight line from A to B. The velocity of P at time t s after leaving A is denoted by v m s-1, where
\(v = 0.04t^3 + ct^2 + kt\).
P takes 5 s to travel from A to B and it reaches B with speed 10 m s-1. The distance AB is 25 m.
A particle moves in a straight line. The displacement of the particle at time t s is s m, where
\(s = t^3 - 6t^2 + 4t\).
Find the velocity of the particle at the instant when its acceleration is zero.
A particle P moves in a straight line. The acceleration \(a \text{ m s}^{-2}\) of P at time \(t\) s is given by \(a = 6t - 12\). The displacement of P from a fixed point O on the line is \(s\) m. It is given that \(s = 5\) when \(t = 1\) and \(s = 1\) when \(t = 3\).
A particle P moves in a straight line from a fixed point O. The velocity v m s-1 of P at time t s is given by
\(v = t^2 - 8t + 12\) for \(0 \leq t \leq 8\).
A particle moves in a straight line. It starts from rest at a fixed point O on the line. Its acceleration at time t s after leaving O is a m s-2, where a = 0.4t^3 - 4.8t^{1/2}.
A particle moves in a straight line. The particle is initially at rest at a point O on the line. At time t s after leaving O, the acceleration a m s-2 of the particle is given by a = 25 - t2 for 0 ≤ t ≤ 9.
\(The acceleration of the particle for t > 9 is given by a = -3t-1/2.\)
A particle moves in a straight line starting from a point O with initial velocity 1 m s-1. The acceleration of the particle at time t s after leaving O is a m s-2, where
\(a = 1.2t^{1/2} - 0.6t\).