At time t=0 a small ball is projected from point A with a...

At time $t=0$ a small ball is projected from point $A$ with a velocity of $200 \mathrm{ft} / \mathrm{sec}$ at the $60^{\circ}$ angle. Neglect atmospheric resistance and determine the two times $t_{1}$ and $t_{2}$ when the velocity of the ball makes an angle of $45^{\circ}$ with the horizontal $x$ -axis.

Key Concepts

Velocity Vector Angle

The velocity vector angle is the angle that the velocity vector makes with the horizontal axis. It provides insight into the direction of the projectile's motion at any given time. By relating the vertical and horizontal components of velocity through the tangent function, one can determine the time at which the projectile’s velocity reaches a particular angle. This concept is key when solving problems that require finding specific instances in the motion where the velocity has a predetermined orientation.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration, such as gravity. These equations link displacement, initial velocity, time, and acceleration, and are used to determine various parameters in projectile motion. They are fundamental to solving problems where the motion needs to be analyzed over time, such as calculating the moments when the projectile’s velocity meets a given condition.

Trigonometry in Kinematics

Trigonometry plays a crucial role in kinematics, especially when dealing with angles associated with the motion of projectiles. By using trigonometric functions like sine, cosine, and tangent, one can relate the components of a vector to its magnitude and direction. In problems where the angle of the velocity vector is specified, the tangent function is particularly useful as it provides the ratio of the vertical component to the horizontal component, allowing for the determination of time or other related quantities.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is influenced only by gravity (assuming air resistance is negligible). In this context, the object follows a curved path called a trajectory. The analysis of projectile motion involves understanding how gravity affects the vertical component of the object's velocity while the horizontal component remains constant, which is critical for solving problems involving time of flight and angles of projection and impact.

Vector Decomposition

Vector decomposition is the process of breaking a vector into its horizontal and vertical components. This technique is essential in projectile motion as it allows the analysis of the object's motion separately in each direction. The initial velocity is split into the horizontal component, which remains constant, and the vertical component, which is affected by gravitational acceleration. This separation simplifies the application of kinematic equations to each direction.

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