Velocity and Constant Acceleration
1.
Two ships P and Q are moving along straight lines with constant velocities. Initially P is at a point O and the position vector of Q relative to O is (6i + 12j) km, where i and j are unit vectors directed due east and due north respectively. The ship P is moving with velocity 10j km h–1 and Q is moving with velocity (−8i + 6j) km h−1. At time t hours the position vectors of P and Q relative to O are p km and q km respectively. (a) (b) (c) Find p and q in terms of t.
(3)
Calculate the distance of Q from P when t = 3.
(3)
Calculate the value of t when Q is due north of P.
(2) (Total 8 marks)
2.
A train starts from rest at a station A and moves along a straight horizontal track. For the first 10 s, the train moves with constant acceleration 1.2 m s–2. For the next 24 s it moves with constant acceleration 0.75 m s–2. It then moves with constant speed for T seconds. Finally it slows down with constant deceleration 3 m s–2 until it comes to rest at a station B. (a) (b) (c) Show that, 34 s after leaving A, the speed of the train is 30 m s–1.
(3)
Sketch a speed-time graph to illustrate the motion of the train as it moves from A to B.
(3)
Find the distance moved by the train during the first 34 s of its journey from A.
(4)
The distance from A to B is 3 km. (d) Find the value of T.
(4) (Total 14 marks)
3.
Two cars A and B are moving in the same direction along a straight horizontal road. At time t = 0, they are side by side, passing a point O on the road. Car A travels at a constant speed of 30 m s–1. Car B passes O with a speed of 20 m s–1, and has constant acceleration of 4 m s–2. Find (a) (b) (c) the speed of B when it has travelled 78 m from O,
(2)
the distance from O of A when B is 78 m from O,
(4)
the time when B overtakes A.
(5) (Total 11 marks)
4.
A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of 1.6 m above the top of the post. (a) Show that
Two ships P and Q are moving along straight lines with constant velocities. Initially P is at a point O and the position vector of Q relative to O is (6i + 12j) km, where i and j are unit vectors directed due east and due north respectively. The ship P is moving with velocity 10j km h–1 and Q is moving with velocity (−8i + 6j) km h−1. At time t hours the position vectors of P and Q relative to O are p km and q km respectively. (a) (b) (c) Find p and q in terms of t.
(3)
Calculate the distance of Q from P when t = 3.
(3)
Calculate the value of t when Q is due north of P.
(2) (Total 8 marks)
2.
A train starts from rest at a station A and moves along a straight horizontal track. For the first 10 s, the train moves with constant acceleration 1.2 m s–2. For the next 24 s it moves with constant acceleration 0.75 m s–2. It then moves with constant speed for T seconds. Finally it slows down with constant deceleration 3 m s–2 until it comes to rest at a station B. (a) (b) (c) Show that, 34 s after leaving A, the speed of the train is 30 m s–1.
(3)
Sketch a speed-time graph to illustrate the motion of the train as it moves from A to B.
(3)
Find the distance moved by the train during the first 34 s of its journey from A.
(4)
The distance from A to B is 3 km. (d) Find the value of T.
(4) (Total 14 marks)
3.
Two cars A and B are moving in the same direction along a straight horizontal road. At time t = 0, they are side by side, passing a point O on the road. Car A travels at a constant speed of 30 m s–1. Car B passes O with a speed of 20 m s–1, and has constant acceleration of 4 m s–2. Find (a) (b) (c) the speed of B when it has travelled 78 m from O,
(2)
the distance from O of A when B is 78 m from O,
(4)
the time when B overtakes A.
(5) (Total 11 marks)
4.
A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of 1.6 m above the top of the post. (a) Show that