Motion in a Plane

Most motions around us, such as a football in flight, do not take place along a straight line. To describe them we need vectors, quantities that have both magnitude and direction. This chapter begins by distinguishing scalars from vectors and explaining position and displacement vectors, equality of vectors and multiplication of a vector by a real number. You will learn to add and subtract vectors graphically using the triangle and parallelogram laws, and analytically by resolving them into rectangular components along the x- and y-axes using the unit vectors i and j. The chapter then describes motion in a plane through position, velocity and acceleration vectors. For constant acceleration the equations v = v0 + at and r = r0 + v0 t + (1/2)at^2 apply, and the motion can be treated as two separate straight-line motions. This idea is used to study projectile motion, deriving its parabolic path, time of flight, maximum height and horizontal range. Finally, you will study uniform circular motion and derive the centripetal acceleration a = v^2/r directed towards the centre.

Detailed Worksheet: Motion in a Plane Section A - Definitions (10 marks) 1. Distinguish between a scalar and a vector with two examples of each. (2 marks) 2. Find the magnitude and the unit vector of A = 3i + 4j. (2 marks) 3. Resolve a force of 10 N acting at 30 degrees to the x-axis into its components. (2 marks) 4. At what angle of projection is the horizontal range maximum? (2 marks) 5. Why is uniform circular motion called accelerated motion? (2 marks) Section B - Calculations and Applications (15 marks) 6. Two forces of 3 N and 4 N act at a point. Find the resultant when the angle between them is (i) 90 degrees (ii) 60 degrees. (3 marks) 7. The position of a particle is r = 3t i + 2t^2 j metres. Find its velocity and acceleration, and its speed at t = 2 s. (3 marks) 8. A ball is thrown horizontally at 10 m/s from the top of a cliff 45 m high. Find the time to reach the ground and the horizontal distance covered. Take g = 10 m/s^2. (3 marks) 9. Show that two angles of projection that add up to 90 degrees give the same horizontal range for the same speed. (3 marks) 10. A stone tied to a string is whirled in a circle of radius 80 cm, making 14 revolutions in 25 s. Find the magnitude and direction of its acceleration. (3 marks) Section C - Diagrams (10 marks) 11. Draw diagrams to explain the triangle law and the parallelogram law of vector addition. (4 marks) 12. Draw the path of a projectile and mark its velocity components at the start, at the highest point and on landing. (3 marks) 13. Draw a diagram for uniform circular motion showing the velocity and acceleration vectors at two points. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Derive the equation of the path, the time of flight, the maximum height and the horizontal range of a projectile launched at angle theta with speed u. (5 marks) 15. A ball is thrown at 20 m/s at 30 degrees above the horizontal. Find its time of flight, maximum height and horizontal range. Take g = 10 m/s^2 and sin 60 = 0.866. (5 marks) 16. Derive the expression for centripetal acceleration in uniform circular motion. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Take g = 9.8 m/s^2 unless stated otherwise. Use i and j for unit vectors along the x- and y-axes.