Momentum
One of those Civil War cannons is fired. The Cannon has a mass of 873kg It fires a 35.0kg cannon ball at a velocity of 145 m/s at an elevation angle of 35.0 degrees. The length of the barrel is 2.10m. (a) What is the recoil velocity of the cannon?(b)What is the KE of the cannon ball as it leaves the cannon? (c)How far does the cannon ball travel in horizontal direction? (d) What is the KE of the cannon ball at the top of its trajectory? (e) What is its momentum at this point? (f) What is the force exerted on the cannon ball?
To answer these questions, we need to use the principles of conservation of momentum, projectile motion, and kinetic energy. Let's break down each question and find the answers step by step:
(a) What is the recoil velocity of the cannon?
To find the recoil velocity of the cannon, we can use the conservation of momentum. The initial momentum of the cannon and cannonball system should be equal to the final momentum.
The initial momentum of the system is given by:
Initial momentum = (Mass of the cannon + Mass of the cannonball) * Recoil velocity (since the cannon was at rest)
As per the question, the cannon mass is 873 kg, and the cannonball mass is 35.0 kg. So the initial momentum would be:
Initial momentum = (873 kg + 35.0 kg) * 0 (recoil velocity is 0 since the cannon starts from rest)
The final momentum of the system is given by:
Final momentum = Mass of the cannonball * Velocity of the cannonball in the opposite direction (recoiling)
Therefore, Final momentum = 35.0 kg * V (recoil velocity of the cannon)
Now we can equate the initial momentum with the final momentum to find the recoil velocity:
(873 kg + 35.0 kg) * 0 = 35.0 kg * V
Simplifying the equation, we find:
0 = 35.0 kg * V
As a result, the recoil velocity of the cannon is 0 m/s.
(b) What is the kinetic energy (KE) of the cannonball as it leaves the cannon?
The kinetic energy is given by the equation:
KE = 0.5 * Mass of the cannonball * (Velocity of the cannonball)^2
Using the given values, the KE of the cannonball can be calculated as follows:
KE = 0.5 * 35.0 kg * (145 m/s)^2
(c) How far does the cannonball travel in the horizontal direction?
To determine the horizontal distance traveled by the cannonball, we need to break down the projectile motion into horizontal and vertical components.
The horizontal component of velocity remains constant throughout the motion, while considering air resistance. Therefore, the horizontal distance can be calculated using the formula:
Distance = Horizontal component of velocity * Time
The horizontal component of velocity, in this case, is given by:
Velocityx = Velocity * cos(angle of elevation)
The time can be evaluated using the formula:
Time = Distance / Horizontal component of velocity
Using the values given:
Velocityx = 145 m/s * cos(35.0 degrees)
Time = 2.10 m / Velocityx
(d) What is the kinetic energy of the cannonball at the top of its trajectory?
At the top of its trajectory, the vertical component of velocity becomes momentarily zero. However, the horizontal velocity component remains unchanged.
The kinetic energy at the top of the trajectory can be calculated using the formula mentioned in part (b), where the velocity is the total velocity:
KE = 0.5 * Mass of the cannonball * (Total velocity)^2
(e) What is its momentum at this point?
To find the momentum of the cannonball at the top of its trajectory, we need to find its velocity in the vertical direction. The vertical velocity can be obtained using the formula:
Velocityy = Velocity * sin(angle of elevation)
The momentum can be calculated by multiplying the mass by the velocity in the vertical direction.
(f) What is the force exerted on the cannonball?
The force exerted on the cannonball is given by Newton's second law of motion. It states that force is equal to the rate of change of momentum. Hence,
Force = (Change in momentum) / (Time taken)
The change in momentum of the cannonball can be calculated by subtracting the initial momentum from the final momentum. The time taken can be derived from the vertical motion of the cannonball.
Using the above information, we can calculate the force exerted on the cannonball.
We'll be glad to critique your work. Surely you can make progress on some of this. (a). Use the horizontal compoment of the cannonball to apply conservation of momentum and get the recoil velocithy.
(b) Use the well known formula for KE
(c) (V^2/g)*sin70
(d) KE at firing minus the KE due to vertical motion. This will equal (M/2)V^2*cos^2 35
(e) M*V*cos35
(f) your turn