A projectile is fired from the point (0,0) with an initial...

A projectile is fired from the point $(0,0)$ with an initial velocity of $v_{0} \mathrm{~m} / \mathrm{s}$ at an angle $\alpha$ above the horizontal. (See Exercise 10.1.58.) If we assume that air resistance is negligible, then the position (in meters) of the projectile after $t$ seconds is given by the parametric equations $$ x=\left(v_{0} \cos \alpha\right) t \quad y=\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2} $$ where $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$ is the acceleration due to gravity. (a) Find the speed of the projectile when it hits the ground. (b) Find the speed of the projectile at its highest point.

A projectile is fired from the point $(0,0)$ with an initial velocity of $v_{0} \mathrm{~m} / \mathrm{s}$ at an angle $\alpha$ above the horizontal. (See Exercise 10.1.58.) If we assume that air resistance is negligible, then the position (in meters) of the projectile after $t$ seconds is given by the parametric equations
$$
x=\left(v_{0} \cos \alpha\right) t \quad y=\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2}
$$
where $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$ is the acceleration due to gravity.
(a) Find the speed of the projectile when it hits the ground.
(b) Find the speed of the projectile at its highest point.

Key Concepts

Projectile Motion

Projectile motion is the two-dimensional motion of an object that is launched into the air and influenced only by gravity (assuming air resistance is negligible). This concept involves analyzing the horizontal and vertical components of an object's movement separately, as gravity only affects the vertical component.

Parametric Equations in Kinematics

Parametric equations are used to describe the motion of an object by expressing its coordinates as functions of an independent parameter, typically time. In the context of projectile motion, these equations detail how the horizontal position and the vertical position vary with time, capturing the effects of initial velocity components and gravitational acceleration.

Decomposition of Velocity

Decomposing velocity involves breaking the initial velocity into two perpendicular components: one in the horizontal direction and one in the vertical direction. This decomposition is essential in projectile motion because it allows the separate analysis of motion along the horizontal axis, which is constant, and the vertical axis, which is affected by gravity.

Effect of Gravity

Gravity is the force that causes acceleration in the vertical direction, acting downward. In projectile motion, this constant acceleration affects the vertical component of velocity, leading to a parabolic trajectory. Gravity is crucial for determining parameters such as the time to reach the highest point, the maximum height, and the speed upon ground impact.

Speed as a Resultant Vector

Speed in projectile motion is determined by calculating the magnitude of the velocity vector, which is obtained by combining the horizontal and vertical components using the Pythagorean theorem. This approach allows for the determination of the projectile's instantaneous speed at any point during its flight, including at the moment of impact or at its peak.

Time of Flight and Velocity at Specific Points

Time of flight refers to the total time a projectile remains in the air before hitting the ground, which is integral to calculating its final speed. Additionally, analyzing the velocity at specific points, such as the highest point (where the vertical velocity component is zero) or the moment of impact, enables the computation of the speed by substituting time into the parametric equations.

Transcript

Please give Ace some feedback