What initial velocity is needed to throw a quarter into a bowl located 0.4 m above the point where it leaves your hand at an angle of 60°?

To determine the initial velocity required to throw a quarter into a bowl that is 0.4 meters above the release point at an angle of 60°, we can use the principles of projectile motion.

When an object is projected at an angle, its motion can be separated into horizontal and vertical components. The vertical motion is influenced by gravity, while the horizontal motion remains constant (ignoring air resistance).

1. Vertical Motion Analysis: The formula for vertical displacement (height) in projectile motion is given by:

[

y = v_{0y} t - \frac{1}{2} g t^2

]

where:

• ( y ) is the vertical displacement (0.4 m in this case),

• ( v_{0y} ) is the initial vertical velocity component,

• ( g ) is the acceleration due to gravity (approximately ( 9.81 , m/s^2 )),

• ( t ) is the time of flight.

The initial vertical velocity component can be calculated as:

[

v_{0y} = v_0 \sin(\theta)

]

where ( \theta ) is the