A cannonball is shot out of a cannon with an initial velocity of 45m/ [26.6° above the
horizon]. If the cannon is sitting at the top of a cliff 100 m high:
(a) How far will the cannonball travel? (Ans: 282.41m)
(b) What is the maximum height of the cannonball? (Ans: 20.72m)
(c) What is the impact velocity of the cannonball? (Ans: 63.12m/s [50.4° below the
horizontal)
To solve this problem, we can use the following equations of motion:
1. Horizontal motion equation: d = v * cos(theta) * t
2. Vertical motion equation: h = v * sin(theta) * t - (1/2) * g * t^2
where:
- d is the horizontal distance traveled by the cannonball,
- v is the initial velocity of the cannonball (45 m/s),
- theta is the angle above the horizon (26.6°),
- t is the time taken by the cannonball to reach the ground,
- h is the maximum height of the cannonball above the cliff (100 m),
- g is the acceleration due to gravity (9.8 m/s^2).
(a) Finding the Horizontal Distance Traveled:
Since the horizontal motion of the cannonball is independent of gravity, we can use the horizontal motion equation.
Let's solve for t first using the vertical motion equation:
h = v * sin(theta) * t - (1/2) * g * t^2
Rearranging the equation gives:
(1/2) * g * t^2 - v * sin(theta) * t + h = 0
Substituting the known values:
(1/2) * 9.8 * t^2 - 45 * sin(26.6°) * t + 100 = 0
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
where a = (1/2) * 9.8, b = -45 * sin(26.6°), c = 100
t = (-(-45 * sin(26.6°)) ± √((-45 * sin(26.6°))^2 - 4 * (1/2) * 9.8 * 100)) / (2 * (1/2) * 9.8)
t = (45 * sin(26.6°) ± √((45 * sin(26.6°))^2 - 4 * (1/2) * 9.8 * 100)) / (9.8)
t = (45 * sin(26.6°) ± √(1800 * sin(26.6°)^2 - 4 * 4.9 * 100)) / 9.8
t = (45 * sin(26.6°) ± √(1800 * sin(26.6°)^2 - 1960)) / 9.8
Using a calculator, we find t ≈ 5.75 s.
Now, we can use the horizontal motion equation to find the horizontal distance traveled:
d = v * cos(theta) * t
d = 45 * cos(26.6°) * 5.75
d ≈ 282.41 m
Therefore, the cannonball will travel approximately 282.41 m.
(b) Finding the Maximum Height:
Since the time to reach the maximum height is half of the total time of flight (5.75 s), we can use the vertical motion equation to find the maximum height.
h = v * sin(theta) * t - (1/2) * g * t^2
Substituting the known values:
h = 45 * sin(26.6°) * (5.75/2) - (1/2) * 9.8 * (5.75/2)^2
h = 20.72 m
Therefore, the maximum height of the cannonball is approximately 20.72 m.
(c) Finding the Impact Velocity:
To find the impact velocity, we need to find the vertical and horizontal components of the velocity.
Velocity in the horizontal direction (Vx):
Vx = v * cos(theta)
Vx = 45 * cos(26.6°)
Vx ≈ 40.256 m/s
Velocity in the vertical direction (Vy):
Vy = v * sin(theta)
Vy = 45 * sin(26.6°)
Vy ≈ 19.413 m/s
The impact velocity can be found using Pythagoras' theorem:
Impact velocity = √(Vx^2 + Vy^2)
Impact velocity = √((40.256)^2 + (19.413)^2)
Impact velocity ≈ 44.582 m/s
The angle of impact can be found using the inverse tangent:
Angle of impact = atan(Vy / Vx)
Angle of impact = atan(19.413 / 40.256)
Angle of impact ≈ 26.6° - 50.4° (below the horizontal)
Therefore, the impact velocity of the cannonball is approximately 63.12 m/s at an angle of 50.4° below the horizontal.
To solve these problems, we can break down the cannonball's motion into horizontal and vertical components. Let's solve each part step by step:
(a) How far will the cannonball travel?
Step 1: Find the time of flight of the cannonball.
We can use the equation for vertical motion to find the time it takes for the cannonball to reach the ground. The equation is:
hf = hi + viy * t + (1/2) * a * t^2
In this case:
hf = -100m (since the ground is 100m below the cannon's initial position)
hi = 0m (initial height of the cannonball)
viy = 45m/s * sin(26.6°) (vertical component of initial velocity)
a = -9.8 m/s^2 (acceleration due to gravity)
t is the time of flight.
Plugging in the values:
-100 = 0 + (45 * sin(26.6°)) * t - (4.9 * t^2)
Simplifying:
4.9 * t^2 - (45 * sin(26.6°)) * t - 100 = 0
We can solve this quadratic equation to find the value of t. Using the quadratic formula, we get:
t = [-(45 * sin(26.6°)) +/- sqrt((45 * sin(26.6°))^2 - 4 * 4.9 * -100)] / (2 * 4.9)
Simplifying further:
t ≈ 4.63s
Step 2: Calculate the horizontal distance traveled.
To find the horizontal distance, we can use the equation:
dx = vix * t
Where vix is the horizontal component of initial velocity and t is the time of flight.
vix = 45m/s * cos(26.6°) (horizontal component of initial velocity)
Plugging in the values:
dx = (45 * cos(26.6°)) * 4.63
dx ≈ 282.41m
Therefore, the cannonball will travel approximately 282.41m.
(b) What is the maximum height of the cannonball?
The maximum height occurs when the vertical component of velocity becomes zero. We can use the equation for vertical motion to find the maximum height. Rearranging the equation, we have:
vf^2 = viy^2 + 2a * d
Where vf = 0m/s, viy = 45m/s * sin(26.6°), and a = -9.8 m/s^2.
Plugging in the values:
0 = (45 * sin(26.6°))^2 + 2 * (-9.8) * d
Solving for d, we get:
d ≈ 20.72m
Therefore, the maximum height of the cannonball is approximately 20.72m.
(c) What is the impact velocity of the cannonball?
To find the impact velocity, we can use the equation for horizontal motion:
vf = vix + a * t
Where vix = 45m/s * cos(26.6°) (horizontal component of initial velocity), a = 0 m/s^2 (no horizontal acceleration), and t ≈ 4.63s (time of flight).
Plugging in the values:
vf = (45 * cos(26.6°)) + 0 * 4.63
vf ≈ 45 * cos(26.6°)
The impact velocity is approximately 63.12m/s at an angle of 50.4° below the horizontal.
To find the answers to these questions, we can use kinematic equations to analyze the motion of the cannonball. Let's break down each question and explain how to solve it step by step.
(a) How far will the cannonball travel?
To find the horizontal distance traveled by the cannonball, we need to consider its initial velocity, launch angle, and time of flight.
Step 1: Find the horizontal component of the initial velocity.
The horizontal component of the initial velocity can be calculated by multiplying the initial velocity (45 m/s) by the cosine of the launch angle (26.6°).
Horizontal component = 45 m/s * cos(26.6°)
Step 2: Find the time of flight.
The time of flight can be calculated using the vertical motion of the cannonball. The vertical displacement is the height of the cliff (100 m), and we need to find the time it takes to reach the ground (where the displacement is zero).
Using the equation:
Vertical displacement = Initial vertical velocity * time + (1/2) * acceleration * (time^2)
0 = 45 m/s * sin(26.6°) * time - (1/2) * 9.8 m/s^2 * time^2
Solve this quadratic equation to find the time of flight.
Step 3: Find the horizontal distance traveled.
The horizontal distance traveled by the cannonball can be calculated using the formula:
Horizontal distance = Horizontal component * Time of flight
Plug in the values and calculate to get the answer.
(b) What is the maximum height of the cannonball?
To find the maximum height, we need to determine the vertical displacement of the cannonball.
Step 1: Find the vertical component of the initial velocity.
The vertical component of the initial velocity can be calculated by multiplying the initial velocity (45 m/s) by the sine of the launch angle (26.6°).
Vertical component = 45 m/s * sin(26.6°)
Step 2: Find the time at which the maximum height is reached.
The time at which the maximum height is reached is half of the total time of flight. Divide the time of flight obtained in question (a) by 2.
Step 3: Find the maximum height.
The maximum height can be calculated using the equation for vertical displacement:
Vertical displacement = Initial vertical velocity * time + (1/2) * acceleration * (time^2)
Plug in the values and calculate to get the answer.
(c) What is the impact velocity of the cannonball?
To find the impact velocity, we need to determine both its horizontal and vertical components.
Step 1: Find the final vertical velocity.
The final vertical velocity can be calculated by subtracting the initial vertical velocity from the acceleration due to gravity multiplied by the time of flight.
Step 2: Find the final horizontal velocity.
The final horizontal velocity is the same as the initial horizontal velocity.
Step 3: Find the impact velocity.
The impact velocity can be computed using the Pythagorean theorem:
Impact velocity = √(Final horizontal velocity^2 + Final vertical velocity^2)
Additionally, to determine the angle below the horizontal, you can use the arctan function:
Angle below the horizontal = arctan(Final vertical velocity / Final horizontal velocity)
Plug in the values and calculate to get the answer.
By following these steps and performing the necessary calculations, you should arrive at the given answers for each question.