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Equations of motion

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For an object moving at constant velocity, the distance it has travelled during a given time interval at a given speed can be calculated using the following formula: distance = speed × time. The instanteneous velocity of an object is its velocity at a specific point in time during its motion, whereas its average velocity is the average of all instantaneous velocities during the object's motion.

The initial velocity, final velocity, acceleration and displacement of an object travelling at a uniform acceleration can be calculated using the following formula: v² − u² = 2as.

Basic equations for uniform acceleration

An object's initial velocity, final velocity, time of motion, displacement and acceleration can all be related using the following formulas: v = u + at, s = (u + v)/2 × t and s = ut + at²/2. The appropriate formula can be used in questions according to the information given in it.

The equations of motion can be derived using the most basic one, s = ut, and the definition of acceleration, a = (v − u)/t. Rearranging the formula of acceleration gives v = u + at. The average velocity of an object can be found by taking the the average of the initial and final velocity, (u + v)/2. Substituting this into the basic equation of motion gives s = (u + v)/2 × t. The third equation of motion, s = ut + (1/2)at², can be found using the area under a velocity-time graph for an object with an initial velocity, u, and final velocity, v and substituting in the formula for acceleration. The fourth equation of motion, v² = u² + 2as, can be found using s = ut. Substituting the average velocity, (u + v)/2, and rearranging for t from the first equation of motion, s = (u + v)/2 × (v − u)/a. Rearranging gives 2as = (u + v)(v − u). Expanding out gives the fourth equation of motion.

An object thrown into the air follows a parabolic path called a trajectory and is called a projectile. This object will move under the influence of gravity. Since gravity acts downwards, it only affects the vertical velocity of the object. Equations of motion can be used to calculate the vertical velocity. There are no horizontal forces acting on the projectile, since air resistance is usually neglected, so the horizontal velocity remains constant during the time of flight and the equation velocity = distance/time can be used. The object travels upwards at a decreasing rate, until its velocity becomes 0 and it starts to accelerate towards the ground until it reaches it.