Variable acceleration and vectors with calculus

A self-driving test vehicle moves along a straight test track. At time t seconds, its displacement from a fixed point O on the track is x metres, where
x = 0.5t³ − 6t² + 18t,   0 ≤ t ≤ 7.
The vehicle is modelled as a particle.

A drone takes off from rest from level ground and rises vertically. At time t seconds after take-off its velocity is v m s⁻¹ upwards, where
v = 2.4t − 0.3t²,   0 ≤ t ≤ 8.
The drone is modelled as a particle.

A model boat moves on a pond. At time t seconds its position vector relative to a fixed point O is r metres, where
r = (3t² − 2t)i + (5 + 4t − t²)j.
The boat is modelled as a particle.

A robot lawnmower moves on a flat lawn. At time t seconds its velocity is v m s⁻¹, where
v = 2ti + (6 − 3t)j.
When t = 0 the lawnmower is at the point with position vector (i − 4j) m relative to a fixed point O. The unit vectors i and j are directed due east and due north respectively. The lawnmower is modelled as a particle.

A sledge is pulled in a straight line across level snow, starting from rest at time t = 0. Figure 1 shows the velocity–time graph of the sledge for 0 ≤ t ≤ 8, where v m s⁻¹ is its velocity at time t seconds. The velocity is modelled by
v = pt + qt²,
where p and q are constants. The sledge is modelled as a particle.

A shuttle moves along a straight horizontal rail. At time t seconds its displacement from a fixed point O on the rail is x metres, where
x = pt² − 0.2t³,   0 ≤ t ≤ 6,
and p is a constant. Figure 1 shows the displacement–time graph of the shuttle. The shuttle is modelled as a particle.

An ice skater moves on a horizontal rink. The unit vectors i and j are directed due east and due north respectively. At time t seconds the position vector of the skater relative to a fixed point O is r metres, where
r = pti + (t² + qt + c)j,   0 ≤ t ≤ 4,
and p, q and c are constants. Figure 1 shows her path on a grid of 1 m squares, with her positions at t = 0, 1, 2, 3 and 4 marked. The skater is modelled as a particle.

A remote-controlled car of mass 1.2 kg moves on a horizontal playground. The unit vectors i and j are directed due east and due north respectively. Figure 1 shows how the i-component and the j-component of the velocity of the car, in m s⁻¹, vary with time t seconds for 0 ≤ t ≤ 4. Both graphs are straight lines. The car is modelled as a particle.

A motorboat moves in a straight line across a calm lake. Its engine is cut at time t = 0. For 0 ≤ t ≤ 10 its velocity is v m s⁻¹, where
v = 8 − 1.6t + 0.08t².
The boat is modelled as a particle.

A test platform of mass 12 kg moves along a vertical guide rail. At time t seconds its displacement above its starting level is y metres, where
y = 0.1t³ − 0.9t² + 2.4t,   0 ≤ t ≤ 5.
The platform is modelled as a particle moving in a vertical line.

A toy train moves along a straight track. It leaves a station O at time t = 0. At time t seconds its velocity is v m s⁻¹, where
v = 0.5t² − 3t + 4,   t ≥ 0.
The train is modelled as a particle.

A remote-controlled boat of mass 3 kg moves on a lake. At time t seconds its velocity is v m s⁻¹, where
v = (t² − 6t + 8)i + (2t + 1)j,   t ≥ 0.
The boat starts from a fixed point O at t = 0. The unit vectors i and j are directed due east and due north respectively. The boat is modelled as a particle.

A delivery drone of mass 1.5 kg flies in a horizontal plane. At time t seconds its position vector relative to a fixed point O is r metres, where
r = (t³ − 3t)i + (t² + 2t)j,   t ≥ 0.
The drone is modelled as a particle.

A cyclist rides along a straight horizontal road. At time t = 0 she passes a point O with speed 2 m s⁻¹. For 0 ≤ t ≤ 8 her acceleration is (1.2 − 0.15t) m s⁻² in the direction of motion. The cyclist is modelled as a particle.

A particle P moves along a straight line. At time t seconds its displacement from a fixed point O on the line is x metres, where
x = pt² − 2t^5/2,   t ≥ 0,
and p is a constant. Figure 1 shows the displacement–time graph of P from t = 0 until P returns to O.

A cart moves along a straight horizontal track in a mine and passes a fixed point O on the track at time t = 0. Figure 1 shows the velocity–time graph of the cart for 0 ≤ t ≤ 7, where v m s⁻¹ is its velocity at time t seconds. The graph is a parabola, and v is a quadratic function of t. The cart is modelled as a particle.

In a shuttle-run drill, a runner starts from rest at time t = 0 and runs in a straight line until she comes to rest at the turning line. At time t seconds her velocity is v m s⁻¹, where
v = pt − qt^3/2,
and p and q are positive constants. Figure 1 shows the velocity–time graph of the runner. The curve passes through the marked point (1, 4), and the tangent to the curve at this point passes through the marked point (3, 10). The runner is modelled as a particle.

A lift is raised vertically by a cable, starting from rest at ground level at time t = 0. Figure 1 shows the velocity–time graph of the lift for 0 ≤ t ≤ 8, where v m s⁻¹ is its upward velocity at time t seconds. In the model,
v = kt² for 0 ≤ t ≤ 2,   v is constant for 2 ≤ t ≤ 6,   v = 3 − k(t − 6)² for 6 ≤ t ≤ 8,
where k is a constant. The lift and its passengers have total mass 800 kg and are modelled as a particle. Take g = 9.8 m s⁻².

A kayak moves on a lake. The unit vectors i and j are directed due east and due north respectively. Figure 1 is a map of part of the lake, with grid squares of side 1 m and a fixed point O. The kayak starts from the point A at time t = 0, and its path passes a buoy at the point B. At time t seconds the velocity of the kayak is v m s⁻¹, where
v = (6 − 2t)i + (2t − 2)j.
The kayak and its paddler have total mass 90 kg and are modelled as a particle.

A kite-surfer moves on the surface of the sea, starting from a fixed point O at time t = 0. The unit vectors i and j are directed due east and due north respectively. At time t seconds the position vector of the kite-surfer relative to O is r metres, where
r = (t² + 2t)i + (6t − t²)j,   0 ≤ t ≤ 5.
Figure 1 shows the path on a grid of 1 m squares, together with an arrow, drawn between two grid points, showing the direction in which the wind is blowing. The kite-surfer is modelled as a particle.

A bead is threaded on a straight horizontal wire. At time t = 0 the bead is at a fixed point O on the wire. At time t seconds its velocity is v m s⁻¹, where
v = 2t² − 8t + 6   for 0 ≤ t ≤ 3,
v = 12 − 108t⁻²   for t > 3.
The bead is modelled as a particle.

A test sled moves along a straight horizontal rail. It starts from rest at a fixed point O on the rail at time t = 0. At time t seconds its acceleration is (pt + q) m s⁻², where p and q are constants. When t = 2 the velocity of the sled is 8 m s⁻¹, and when t = 4 the sled is again instantaneously at rest. The sled is modelled as a particle.

A small hovercraft of mass 0.5 kg moves on a large flat sheet of ice. At time t seconds the resultant force on it is F newtons, where
F = (3t − 2)i − j.
When t = 0 the hovercraft has velocity (2i + 6j) m s⁻¹ and is at the point with position vector (−i + 3j) m relative to a fixed point O. The unit vectors i and j are directed due east and due north respectively. The hovercraft is modelled as a particle.

A jet-ski of mass 250 kg moves on the sea. At time t seconds its velocity is v m s⁻¹, where
v = (t² − 5t + 6)i + (t² − 4)j,   t ≥ 0.
The unit vectors i and j are directed due east and due north respectively. The jet-ski is modelled as a particle.

A metro train travels along a straight horizontal track between two stations. It starts from rest at the first station at time t = 0 and comes to rest at the second station. At time t seconds its velocity is v m s⁻¹, where
v = kt²(60 − t),
and k is a constant. Figure 1 shows the velocity–time graph of the train, with two marked points. The train has mass 1.2 × 10⁵ kg and is modelled as a particle.

A small inspection probe moves along a straight horizontal pipe. At time t = 0 it passes a fixed point O in the pipe with speed 5 m s⁻¹ in the negative direction. At time t seconds its velocity is v m s⁻¹ and its displacement from O is x metres. Figure 1 shows the acceleration–time graph of the probe for 0 ≤ t ≤ 6, which is a straight line. The probe is modelled as a particle.

Two model cars, P and Q, move on a horizontal car park. The unit vectors i and j are directed due east and due north respectively. Figure 1 is a plan with grid squares of side 1 m and a fixed point O. It shows the starting points, A of P and B of Q, at time t = 0, and the paths of the cars for 0 ≤ t ≤ 2.3. At time t seconds the velocities of P and Q are
v_P = (3t² − 2)i + 4tj m s⁻¹   and   v_Q = −2ti − j m s⁻¹.
The cars are modelled as particles.

A particle of mass 2 kg moves on a smooth horizontal plane. The unit vectors i and j are directed due east and due north respectively. At time t seconds the acceleration of the particle is a m s⁻², where
a = pti + qj,
and p and q are constants. Figure 1 is a velocity diagram, drawn on a grid of unit squares, showing the velocity of the particle at t = 0 and at t = 2. The particle is at a fixed point O when t = 0.