Projectile Motion and Velocity

Projectile motion is the motion of an object thrown into the air at an angle to the horizontal. The object follows a curved path, known as a trajectory, due to the influence of gravity. The velocity of the projectile can be broken down into two components: the horizontal component and the vertical component.

The horizontal component of velocity remains constant throughout the motion of the projectile. This is because there is no horizontal force acting on the projectile, so it continues to move at a constant speed in the horizontal direction.

The vertical component of velocity changes due to the influence of gravity. As the projectile moves upward, the vertical component of velocity decreases until it reaches its maximum height. At this point, the vertical component of velocity becomes zero. Then, as the projectile moves downward, the vertical component of velocity increases in magnitude but in the opposite direction.

Initial Velocity and Angle

In this question, the projectile is thrown with an initial speed u at an angle of 60 degrees with the horizontal. This means that the initial velocity can be broken down into its horizontal and vertical components.

The horizontal component of velocity can be calculated using the formula:

v_x = u * cos(theta)

Where v_x is the horizontal component of velocity, u is the initial speed, and theta is the angle with respect to the horizontal.

The vertical component of velocity can be calculated using the formula:

v_y = u * sin(theta)

Where v_y is the vertical component of velocity, u is the initial speed, and theta is the angle with respect to the horizontal.

Velocity Becoming Perpendicular

In projectile motion, the velocity of the projectile becomes perpendicular to the initial velocity at two points in its trajectory: at the highest point and at the same height on the way down.

At the highest point of the trajectory, the vertical component of velocity becomes zero while the horizontal component of velocity remains constant. This means that the velocity vector is purely horizontal and perpendicular to the initial velocity vector.

On the way down, when the projectile reaches the same height as the starting point, the vertical component of velocity is the same magnitude as the initial vertical component of velocity but in the opposite direction. The horizontal component of velocity remains constant. This means that the velocity vector is again purely horizontal and perpendicular to the initial velocity vector.

Therefore, the height of the object when its velocity becomes perpendicular to the initial velocity is equal to the initial height of the object.