A cannon ball is projected so as to attain a maximum range. Find the maximum height attained if the initial velocity is U.
π/4 is the launch angle for maximum range on a level surface
the vertical component of the launch velocity would be ... U √2 / 2
kinetic energy at launch is potential energy at peak ... U^2 / 4 = g h
solve for h
Need answer
To find the maximum height attained by a cannonball projected with an initial velocity U, we need to consider the projectile motion.
Let's break down the problem step by step:
Step 1: Split the motion into horizontal and vertical components:
- The horizontal component of the motion has a constant velocity, as there is no acceleration in that direction.
- The vertical component is affected by gravity and has an initial velocity of U.
Step 2: Find the time it takes for the cannonball to reach its maximum height:
- The time taken to reach the maximum height is half of the total flight time, as the ascent time is equal to the descent time.
- The total flight time can be found using the equation: t = 2U/g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Therefore, the time taken to reach the maximum height is: t/2 = U/(2g).
Step 3: Calculate the maximum height attained:
- The maximum height reached can be found using the formula: H = (U^2)(sin^2θ) / (2g), where θ is the angle of projection.
- In this case, we want to find the maximum height reached, so we need to consider the angle that gives the maximum range.
- The angle for maximum range is 45 degrees, as it will result in the same horizontal and vertical components.
- Plugging in this value, the formula becomes: H = (U^2)/(4g).
So, the maximum height attained by the cannonball is (U^2)/(4g).
To find the maximum height attained by a cannonball projected with an initial velocity U to achieve maximum range, we can use the concept of projectile motion.
Projectile motion can be analyzed by breaking it into two independent components: horizontal motion and vertical motion.
The horizontal motion is unaffected by gravity, and the only force acting is the initial velocity U. Hence, the horizontal velocity remains constant throughout the motion.
The vertical motion is influenced by gravity, which acts to slow down the cannonball in the upward direction and speed it up in the downward direction.
Breaking down the problem further, we know that the time taken for the cannonball to reach its maximum height and return to the same vertical height is the same. Let's call this time "t".
During the ascent (upward motion), the velocity decreases until it reaches zero at the peak, and then it starts to increase in the downward motion due to the acceleration by gravity.
To find the maximum height, we need to determine the time of flight and use it to calculate the height at that time.
Step 1: Find the time of flight.
The time of flight can be calculated using the following formula:
t = (2 * U * sin(θ)) / g
where U is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity (approximately 9.8 m/s²).
Step 2: Calculate the maximum height.
The maximum height can be calculated using the formula:
H = (U² * sin²(θ)) / (2 * g)
where U is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.
Note: If the angle of projection is not given, you can assume it to be 45 degrees, as it generally yields the maximum range for a projectile.
By substituting the appropriate values for U and θ, you can calculate the maximum height attained by the cannonball.