What is the maximum height above ground a projectile of mass 0.67 kg, launched from ground level, can achieve if you are able to give it an initial speed of 78.5 m/s?
V^2 = Vo^2 + 2g*h
V = 0
Vo = 78.5 m/s.
g = -9.8m/s^2
Solve for h.
To determine the maximum height above ground a projectile can achieve, we can use the principles of projectile motion and the laws of conservation of energy.
Here's how we can approach this problem step by step:
Step 1: Identify the given values:
- Mass of the projectile, m = 0.67 kg
- Initial speed of the projectile, u = 78.5 m/s
Step 2: Determine the gravitational potential energy at the maximum height.
At maximum height, the projectile momentarily comes to a stop before falling back down. At this point, all of its kinetic energy is converted into gravitational potential energy. Therefore, the maximum height can be determined by equating the initial kinetic energy to the final gravitational potential energy.
The formula for kinetic energy is: KE = (1/2) * m * v^2
Where:
KE = Kinetic Energy
m = Mass of the projectile
v = Velocity of the projectile
The formula for gravitational potential energy is: PE = m * g * h
Where:
PE = Gravitational Potential Energy
m = Mass of the projectile
g = Acceleration due to gravity (approximately 9.8 m/s^2)
h = Height above the reference point (initial height)
Step 3: Apply conservation of energy.
At maximum height, all of the initial kinetic energy is converted into potential energy. Therefore, we can set the initial kinetic energy equal to the gravitational potential energy at maximum height and solve for h.
(1/2) * m * u^2 = m * g * h
Step 4: Solve for maximum height, h.
Plug in the given values into the equation:
(1/2) * 0.67 kg * (78.5 m/s)^2 = 0.67 kg * 9.8 m/s^2 * h
Simplify the equation:
(0.5) * 0.67 kg * (78.5 m/s)^2 = 0.67 kg * 9.8 m/s^2 * h
Calculate the left side of the equation:
0.5 * 0.67 kg * (78.5 m/s)^2 = 0.5 * 0.67 kg * 6162.25 m^2/s^2 = 2060.7025 J
Solve for h:
2060.7025 J = 0.67 kg * 9.8 m/s^2 * h
Divide both sides by (0.67 kg * 9.8 m/s^2):
h = 2060.7025 J / (0.67 kg * 9.8 m/s^2)
Calculating h:
h ≈ 315.015 m
So, the maximum height above ground that the projectile can achieve is approximately 315.015 meters.