4. A grapefruit is launched out of a cannon at 17 m/s at 25° above horizontal. What is the time inthe air if the cannon was 5 m above the ground?
Question
- A grapefruit is launched out of a cannon at 17 m/s at 25° above horizontal. What is the time inthe air if the cannon was 5 m above the ground?
Solution 1
To find the time in the air, we can use the kinematic equations of motion.
Step 1: Resolve the initial velocity into its horizontal and vertical components.
The initial velocity of the grapefruit is given as 17 m/s at 25° above horizontal.
The horizontal component of the velocity (Vx) can be found using the formula: Vx = V * cos(θ), where V is the magnitude of the velocity and θ is the angle above horizontal.
Vx = 17 m/s * cos(25°) = 15.38 m/s (rounded to two decimal places)
The vertical component of the velocity (Vy) can be found using the formula: Vy = V * sin(θ).
Vy = 17 m/s * sin(25°) = 7.26 m/s (rounded to two decimal places)
Step 2: Determine the time taken for the grapefruit to reach its maximum height.
The time taken to reach the maximum height can be found using the formula: t = Vy / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
t = 7.26 m/s / 9.8 m/s^2 = 0.74 seconds (rounded to two decimal places)
Step 3: Determine the total time in the air.
Since the grapefruit reaches its maximum height halfway through its flight, the total time in the air is twice the time taken to reach the maximum height.
Total time in the air = 2 * 0.74 seconds = 1.48 seconds (rounded to two decimal places)
Therefore, the time in the air for the grapefruit is approximately 1.48 seconds.
Solution 2
To find the time in the air, we can use the kinematic equations of motion.
Step 1: Resolve the initial velocity into its horizontal and vertical components.
The initial velocity of the grapefruit is given as 17 m/s at 25° above horizontal.
The horizontal component of the velocity (Vx) can be found using the formula: Vx = V * cos(θ), where V is the magnitude of the velocity and θ is the angle above horizontal.
Vx = 17 m/s * cos(25°) = 15.38 m/s (rounded to two decimal places)
The vertical component of the velocity (Vy) can be found using the formula: Vy = V * sin(θ).
Vy = 17 m/s * sin(25°) = 7.26 m/s (rounded to two decimal places)
Step 2: Determine the time taken for the grapefruit to reach its maximum height.
The time taken to reach the maximum height can be found using the formula: t = Vy / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
t = 7.26 m/s / 9.8 m/s^2 = 0.74 seconds (rounded to two decimal places)
Step 3: Determine the total time in the air.
Since the grapefruit reaches its maximum height halfway through its flight, the total time in the air is twice the time taken to reach the maximum height.
Total time in the air = 2 * 0.74 seconds = 1.48 seconds (rounded to two decimal places)
Therefore, the time in the air for the grapefruit is approximately 1.48 seconds.
Solution 3
To find the time in the air, we can use the kinematic equations of motion.
Step 1: Resolve the initial velocity into its horizontal and vertical components.
The initial velocity of the grapefruit is given as 17 m/s at 25° above horizontal.
The horizontal component of the velocity (Vx) can be found using the formula: Vx = V * cos(theta), where V is the magnitude of the velocity and theta is the angle above horizontal.
Vx = 17 m/s * cos(25°) = 15.38 m/s (rounded to two decimal places)
The vertical component of the velocity (Vy) can be found using the formula: Vy = V * sin(theta), where V is the magnitude of the velocity and theta is the angle above horizontal.
Vy = 17 m/s * sin(25°) = 7.26 m/s (rounded to two decimal places)
Step 2: Determine the time taken for the grapefruit to reach its maximum height.
The time taken to reach the maximum height can be found using the formula: t = Vy / g, where Vy is the vertical component of the velocity and g is the acceleration due to gravity (approximately 9.8 m/s^2).
t = 7.26 m/s / 9.8 m/s^2 = 0.74 seconds (rounded to two decimal places)
Step 3: Determine the total time in the air.
Since the grapefruit reaches its maximum height halfway through its flight, the total time in the air is twice the time taken to reach the maximum height.
Total time in the air = 2 * 0.74 seconds = 1.48 seconds (rounded to two decimal places)
Therefore, the time in the air for the grapefruit is approximately 1.48 seconds.
Solution 4
To find the time in the air, we can use the kinematic equations of motion.
Step 1: Resolve the initial velocity into its horizontal and vertical components.
The initial velocity of the grapefruit is given as 17 m/s at 25° above horizontal.
The horizontal component of the velocity (Vx) can be found using the formula: Vx = V * cos(θ), where V is the magnitude of the velocity and θ is the angle above horizontal.
Vx = 17 m/s * cos(25°) = 15.38 m/s (rounded to two decimal places)
The vertical component of the velocity (Vy) can be found using the formula: Vy = V * sin(θ).
Vy = 17 m/s * sin(25°) = 7.26 m/s (rounded to two decimal places)
Step 2: Determine the time taken for the grapefruit to reach its maximum height.
The time taken to reach the maximum height can be found using the formula: t = Vy / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
t = 7.26 m/s / 9.8 m/s^2 = 0.74 seconds (rounded to two decimal places)
Step 3: Determine the total time in the air.
Since the grapefruit reaches its maximum height halfway through its flight, the total time in the air is twice the time taken to reach the maximum height.
Total time in the air = 2 * 0.74 seconds = 1.48 seconds (rounded to two decimal places)
Therefore, the time in the air for the grapefruit is approximately 1.48 seconds.
Solution 5
To find the time in the air, we can use the kinematic equations of motion.
Step 1: Resolve the initial velocity into its horizontal and vertical components.
The initial velocity of the grapefruit is given as 17 m/s at 25° above horizontal.
The horizontal component of the velocity (Vx) can be found using the formula: Vx = V * cos(θ), where V is the magnitude of the velocity and θ is the angle above horizontal.
Vx = 17 m/s * cos(25°) = 15.38 m/s (rounded to two decimal places)
The vertical component of the velocity (Vy) can be found using the formula: Vy = V * sin(θ).
Vy = 17 m/s * sin(25°) = 7.26 m/s (rounded to two decimal places)
Step 2: Determine the time taken for the grapefruit to reach its maximum height.
The time taken to reach the maximum height can be found using the formula: t = Vy / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
t = 7.26 m/s / 9.8 m/s^2 = 0.74 seconds (rounded to two decimal places)
Step 3: Determine the total time in the air.
Since the grapefruit reaches its maximum height halfway through its flight, the total time in the air is twice the time taken to reach the maximum height.
Total time in the air = 2 * 0.74 seconds = 1.48 seconds (rounded to two decimal places)
Therefore, the time in the air for the grapefruit is approximately 1.48 seconds.
Solution 6
To find the time in the air, we can use the kinematic equations of motion.
Step 1: Resolve the initial velocity into its horizontal and vertical components. The initial velocity of the grapefruit is 17 m/s at an angle of 25° above the horizontal. The horizontal component of the velocity (Vx) can be found using the equation Vx = V * cos(theta), where V is the magnitude of the velocity and theta is the angle. Vx = 17 m/s * cos(25°) = 15.38 m/s. The vertical component of the velocity (Vy) can be found using the equation Vy = V * sin(theta). Vy = 17 m/s * sin(25°) = 7.21 m/s.
Step 2: Determine the time it takes for the grapefruit to reach its maximum height. The vertical motion of the grapefruit can be analyzed using the equation: Vy = Vy0 + a * t, where Vy0 is the initial vertical velocity, a is the acceleration (which is -9.8 m/s^2 due to gravity), and t is the time. At the maximum height, the vertical velocity becomes zero, so we can rearrange the equation to solve for the time: 0 = 7.21 m/s - 9.8 m/s^2 * t. Solving for t, we get: t = 7.21 m/s / 9.8 m/s^2 = 0.736 seconds.
Step 3: Determine the total time of flight. Since the time it takes for the grapefruit to reach its maximum height is the same as the time it takes to fall back down to the ground, the total time of flight is twice the time it takes to reach the maximum height. Total time of flight = 2 * 0.736 seconds = 1.472 seconds.
Therefore, the time in the air for the grapefruit is approximately 1.472 seconds.
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