A cannon ball is projected so as to attain maximum range. Find the maximum height attained if the initial velocity is u
To find the maximum height attained by a cannonball projected for maximum range, we will use the concept of projectile motion.
Let's analyze the motion of the cannonball step by step:
Step 1: Split the motion into horizontal and vertical components.
Since the cannonball is projected for maximum range, we can assume that the launch angle is 45 degrees (the optimum angle for maximum range). At this angle, the initial velocity of the cannonball can be split into two components: horizontal (ux) and vertical (uy).
Horizontal component: ux = u * cos(45°)
Vertical component: uy = u * sin(45°)
Step 2: Analyze the vertical motion.
In vertical motion, the only force acting on the cannonball is gravity, which causes it to move in a downward direction.
The equation of motion for vertical motion is:
h = uy * t - (1/2) * g * t^2
Where:
h = height
uy = vertical component of the initial velocity (uy = u * sin(45°))
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time of flight (the time it takes for the cannonball to hit the ground)
Step 3: Determine the time of flight.
Since the cannonball is projected for maximum range, the time of flight can be found using the formula:
t = (2 * uy) / g
Step 4: Substitute the time of flight into the equation of motion.
Substituting the value of time (t) into the equation of motion for vertical motion, we can find the maximum height (h) attained by the cannonball.
h = (u * sin(45°)) * [(2 * u * sin(45°)) / g] - (1/2) * g * [(2 * u * sin(45°)) / g]^2
Simplifying the equation gives us:
h = (u^2 * sin^2(45°)) / (2 * g)
Therefore, the maximum height attained by the cannonball projected for maximum range is given by:
h = (u^2) / (2 * g)
To find the maximum height attained by a cannonball when it is projected to achieve maximum range, we can use the principles of projectile motion.
Let's break down the problem step-by-step:
Step 1: Analyze the motion of the cannonball
The motion of the cannonball can be divided into two components: horizontal and vertical. The vertical component is influenced by gravity, while the horizontal component remains constant.
Step 2: Determine the maximum height
At the maximum height, the vertical component of the cannonball's velocity becomes zero. From this point, the cannonball starts descending back towards the ground.
Step 3: Apply the equations of motion
Using the equations of motion, we can calculate the time taken to reach the maximum height and the maximum height itself.
Equation for the vertical component of velocity:
v = u - gt
Where:
v = final vertical velocity (zero at max height)
u = initial vertical velocity (which is the vertical component of the initial velocity, given as u * sin(angle))
g = acceleration due to gravity (-9.8 m/s^2)
t = time taken
Step 4: Calculate the time taken to reach maximum height
Since the vertical component of the velocity becomes zero at the maximum height, we can rearrange the equation as follows:
0 = u - gt
Solving for t:
t = u / g
Step 5: Calculate the maximum height
To find the maximum height, we need to calculate the displacement. The displacement (s) can be calculated using the equation:
s = ut + (1/2)gt^2
However, at maximum height, the displacement is zero:
0 = ut + (1/2)gt^2
Simplifying the equation yields:
t = 2u / g
Substituting this value back into the equation for time:
t = u / g
Therefore, the maximum height (s) can be calculated as:
s = ut - (1/2)gt^2
Substituting t = u / g:
s = u(u / g) - (1/2)g(u / g)^2
s = (u^2 / g) - (u^2 / 2g)
s = (2u^2 - u^2) / 2g
s = u^2 / 2g
Hence, the maximum height attained by the cannonball is u^2 / 2g.