A 4 kg projectile has 1400 J of kinetic energy when it leaves the ground. If it has 120 J of kinetic energy at its highest point, how high did it travel?
KE = 0.5*M*Vo^2 = 1400 J.
Vo^2 = 1400/0.5M = 1400/2 = 700
Vo = 26.5 m/s. = Initial velocity.
KE = 0.5*M*Xo^2 = 120 J.
Xo^2 = 120/0.5M = 120/2 = 60
Xo = 7.75 m/s=Hor. component of initial
velocity.
Xo = Vo*Cos A = 7.75 m/s.
26.5*Cos A = 7.75
Cos A = 0.29230
A = 73o
Yo = Vo*sin A = 26.5*sin 73 = 25.3=Ver.
component of initial velocity.
Y^2 = Yo^2 + 2g*h = 0
h = -(Yo^2)/2g = -(25.3^2)/-19.6=32.8 m.
To determine the height the projectile traveled, we need to use the principle of conservation of energy. The total mechanical energy of the projectile remains constant throughout its motion, which includes both its potential energy and kinetic energy.
We are given that the projectile has 1400 J of kinetic energy when it leaves the ground and 120 J at its highest point. The difference between these two kinetic energies represents the change in potential energy.
The potential energy at the highest point is given by the formula:
Potential Energy = Mass × Gravitational Acceleration × Height
Let's denote the mass of the projectile as m, the gravitational acceleration as g, and the height it reaches as h.
At the ground level, the projectile has no potential energy but only kinetic energy:
Kinetic Energy at the ground = 1400 J
At the highest point, the projectile has no kinetic energy but only potential energy:
Potential Energy at the highest point = Mass × Gravitational Acceleration × Height = mgh
The change in potential energy can be calculated as:
Change in Potential Energy = Potential Energy at the highest point - Potential Energy at the ground
= mgh - 0
Since the projectile has 120 J of kinetic energy at its highest point, the change in potential energy is equal to 120 J:
120 J = mgh - 0
We are given that the mass of the projectile is 4 kg. The gravitational acceleration can be considered approximately as 9.8 m/s^2.
Substituting the given values into the equation:
120 J = (4 kg) × (9.8 m/s^2) × h
Simplifying the equation:
120 J = 39.2 h
Dividing both sides of the equation by 39.2:
h = 120 J / 39.2
h ≈ 3.06 meters
Therefore, the projectile traveled approximately 3.06 meters in height.