The height y and the distance x along the horizontal planc...

The height $y$ and the distance $x$ along the horizontal planc of a projectile on a certain planct (assuming flat surfacc) with no surrounding atmosphere are given by $x=6 t$ and $y=8 t-5 t^{2}$, wherc $x$ and $v$ are in metre and time $t$ is in second. Find the velocity with which the body is projected, the maximum height attained and the range of the projectile. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$

Key Concepts

Projectile Motion

Projectile motion describes the motion of an object that is launched into the air and then moves under the influence of gravity alone. This concept highlights that the object's horizontal motion is at constant velocity while its vertical motion is accelerated by gravity, leading to a parabolic trajectory.

Kinematics in Two Dimensions

Kinematics in two dimensions involves analyzing motion along two perpendicular axes—typically horizontal and vertical. By separating the motion into these components, one can apply the appropriate kinematic equations: the horizontal component, with constant velocity, and the vertical component, with constant acceleration (gravity), are addressed independently.

Initial Velocity Calculation

The initial velocity of a projectile is a vector quantity that can be broken down into horizontal and vertical components. Its magnitude is determined by combining these components using the Pythagorean theorem. This calculation is essential in determining the overall speed and direction of the projectile at the moment of launch.

Maximum Height Determination

The maximum height in projectile motion is the highest vertical position reached and occurs when the vertical component of the velocity becomes zero. This concept utilizes the kinematic equations to determine the time at which this occurs and then computes the corresponding vertical displacement.

Range of a Projectile

The range of a projectile is defined as the total horizontal distance traveled during its flight. It is calculated by determining the time of flight—found by setting the vertical displacement equal to zero once the projectile has landed—and then multiplying this time by the constant horizontal velocity.

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