Motion, Equations of Motion and Projectiles

What you need to know

• Distance is the total path length travelled and is a scalar; displacement is the straight-line change of position from start to finish and is a vector. A boy who runs once round a 400 m track covers 400 m distance but zero displacement.
• Speed is distance per unit time; velocity is displacement per unit time. Average speed over a whole journey is total distance divided by total time, not the average of the separate speeds.
• Acceleration is the rate of change of velocity. It is positive when velocity increases, negative (retardation or deceleration) when velocity decreases, and it exists even at constant speed if the direction is changing, as in circular motion.
• The five equations of uniformly accelerated motion are v = u + at, s = ut + (1/2)at^2, v^2 = u^2 + 2as, s = ((u + v)/2)t and, for the distance in the nth second, s_n = u + (a/2)(2n - 1). Choose the one that contains only the quantity you want and the quantities you were given.
• On a displacement-time graph the slope is velocity. A straight line means uniform velocity; a curve bending upwards means acceleration; a horizontal line means the body is at rest.
• On a velocity-time graph the slope is acceleration and the area under the graph is the distance covered. Split the area into triangles and rectangles and add them; this is how most WAEC graph questions are scored.
• A body falling freely from rest has u = 0 and a = g = 10 m/s^2 downward (WAEC uses g = 10 m/s^2 unless the question states otherwise). A body thrown vertically upward has a = -10 m/s^2 and its velocity is zero at the highest point, though its acceleration there is still 10 m/s^2 downward.
• For vertical projection, time to reach maximum height is t = u/g, maximum height is H = u^2/(2g), and total time of flight back to the starting level is 2u/g. The speed of return to the launch level equals the launch speed.
• A projectile launched at angle theta has two independent motions: horizontal motion at constant velocity u cos theta, because there is no horizontal force, and vertical motion under gravity with initial vertical velocity u sin theta.
• For a projectile on level ground: time of flight T = 2u sin(theta)/g, maximum height H = u^2 sin^2(theta)/(2g), and range R = u^2 sin(2 theta)/g. Maximum range occurs at theta = 45 degrees, and two angles that add to 90 degrees (for example 30 and 60) give the same range.
• For a body projected horizontally from a height h, such as a bomb released from an aircraft, the initial vertical velocity is zero, so h = (1/2)g t^2 gives the time of fall and the horizontal range is R = u t.
• In circular motion the body moves with constant speed but changing velocity, so it accelerates towards the centre. Centripetal acceleration is v^2/r and the force providing it is mv^2/r, directed inward; there is no outward force acting on the body itself.
• Relative velocity is found by vector subtraction: the velocity of A relative to B is vA - vB. Two cars at 60 km/h approaching each other close at 120 km/h; moving in the same direction they close at zero.
• A ticker-tape timer vibrating at 50 Hz makes one dot every 0.02 s, so the spacing of the dots measures velocity directly and the change in spacing measures acceleration. This is the standard practical for uniform acceleration.