The total time during which the projectile is in flight is given by \( T_{ f } = 2 ( v_{ o } sin{ theta_{ o } } ) / g \).

The sources define uniform circular motion as an object following a circular path at a constant speed.

The sources state that a vector quantity must have both magnitude and direction and obey the triangle law of addition or the parallelogram law of addition.

By definition, the direction of average acceleration is the same as the direction of the change in velocity vector \( Delta mathbf{ v } \).

At the point of maximum height, the vertical component of velocity \( v_{ y } \) is zero.

The unit vectors used to specify directions along the coordinate axes are denoted by \( hat{ i } \), \( hat{ j } \), and \( hat{ k } \).

The acceleration directed towards the centre is called centripetal acceleration, a term proposed by Newton.

Linear velocity is related to angular speed by the equation \( v = omega R \).

The direction of velocity at any point on the path of an object is tangential to the path at that point and in the direction of motion.

The sources demonstrate that the equation of the path of a projectile is that of a parabola.

Multiplying a vector \( mathbf{ A } \) by a negative number gives a vector whose direction is opposite to the direction of \( mathbf{ A } \).

The sources state that kinematic equations for uniform acceleration do not apply to uniform circular motion because the direction of acceleration is changing.

The sources provide the formula for the magnitude of centripetal acceleration as \( a_{ c } = v^{ 2 } / R \).

The sources state that the range \( R \) is maximum when \( sin{ 2theta_{ o } } \) is maximum, which occurs at \( 45^{ circ } \).

The sources state that vector addition is commutative, meaning the order of addition does not change the resultant vector.

Ranges are equal for elevations which exceed or fall short of \( 45^{ circ } \) by equal amounts.

The angular speed \( omega \) is defined as the time rate of change of angular displacement.

Using simple trigonometry, the sources show that the \( y \)-component is \( A_{ y } = A sin{ theta } \).

The magnitude of the resultant is calculated using the Law of Cosines: \( R = sqrt{ A^{ 2 } + B^{ 2 } + 2AB cos{ theta } } \).

According to the sources, a scalar quantity is a quantity with magnitude only, such as mass or temperature.

The sources define a projectile as an object that is in flight after being thrown or projected, such as a football or cricket ball.

Galileo first stated the independency of the horizontal and vertical components of projectile motion in 1632.

The sources define a null vector as having zero magnitude, and because of this, its direction cannot be specified.

The sources refer to the procedure of arranging vectors head-to-tail to find the resultant as the head-to-tail method or triangle method.

The acceleration in uniform circular motion is always directed towards the centre of the circle.