Question

Write formulas for the initial speed of a projectile you will use in this lab using both kinematics and energy methods (two formulas). The kinematics formula should be in terms of horizontal and vertical displacements. The energy formula should be in terms or vertical height, projectile mass, and solid pendulum mass. Identify all variables and SI units.

Answer 1

The initial speed of a projectile using kinematics can be found using separate equations for horizontal and vertical motion, such as x = xo + Vxt and y = yo + voyt - 1/2gt². Energy methods require using conservation of energy, with the formula PE_final = KE_initial - PE_initial.

Step-by-step explanation:

To calculate the initial speed of a projectile using kinematics, we start by analyzing the horizontal and vertical components of the motion separately. For horizontal motion, we assume a constant velocity and use the equation x = xo + Vxt, where x is the horizontal displacement (m), xo is the initial horizontal position which is typically zero (m), Vx is the horizontal velocity (m/s), and t is the time (s). For vertical motion, we use the equation y = yo + voyt - 1/2gt², where y is the vertical displacement (m), yo is the initial vertical position (m), voy is the initial vertical velocity (m/s), g is the acceleration due to gravity (9.80 m/s²), and t is again the time (s).

Using energy methods to find initial speed involves conservation of mechanical energy. The formula is KE_initial + PE_initial = KE_final + PE_final, where KE is kinetic energy and PE is potential energy. Assuming we are calculating the initial speed at launch, and taking the reference point for zero gravitational potential energy at the launch position, the energy formula can be rearranged to PE_final = KE_initial - PE_initial when the projectile is at a maximum height, h. Since KE_initial is the initial kinetic energy, PE_final is the final potential energy at height h, which equals mgh, m is projectile mass (kg), and g is acceleration due to gravity (9.80 m/s²).

Answer 2

x = v₀ cos θ t, y = v₀ sin θ t - ½ g t² , y = ½ v² / g

Step-by-step explanation:

The formulas for launching projectiles is

as in the x axis there is no acceleration

x = v₀ₓ t

x = v₀ cos θ t

on the axis and the acceleration is the acceleration of gravity

y =
`v_(oy)` t - ½ g t²

y = v₀ sin θ t - ½ g t²

the formulas for the position with energy metomentodos, we must see that since there is no friction the mechanical energy is conserved, we write the energy for two points

initial. Highest point

`Em_(o)` = U = m g y

final . Lowest point

Em_{f} = K = ½ m v²

Em_{o} = Em_{f}

mg y = ½ m v²

y = ½ v² / g