A cannon placed at the top of a 100.0 m high cliff fires horizontally to a platoon of Hussars situated initially 1.0 km away from the base of the cliff. At what initial velocity should the cannon shell be fired assuming the hussars are moving with a constant speed of 10 m/s towards the cliff?
To determine the initial velocity at which the cannon shell should be fired, we can use the concept of relative motion.
Let's break down the problem step-by-step:
Step 1: Determine the time it takes for the cannonball to reach the hussars.
The horizontal distance between the cannon and the hussars is 1.0 km or 1000 m. The cannonball will take the same time to reach the hussars as it would take to fall vertically from the height of the cliff (100.0 m).
Using the formula for vertical free fall: h = (1/2)gt^2, where h is the height, g is the acceleration due to gravity (9.8 m/s^2), and t is the time taken, we can solve for t:
100.0 = (1/2) * 9.8 * t^2
Simplifying the equation:
200.0 = 9.8t^2
t^2 = 200.0 / 9.8
t^2 ≈ 20.41
t ≈ √(20.41)
t ≈ 4.52 seconds
Step 2: Determine the horizontal distance the hussars travel during this time.
The hussars are moving towards the cliff with a constant speed of 10 m/s for 4.52 seconds. So, the distance they travel is:
Distance = Speed * Time
Distance = 10 * 4.52
Distance ≈ 45.2 meters
Step 3: Determine the total horizontal distance the cannonball needs to travel.
The total horizontal distance the cannonball needs to travel is the initial distance between the cannon and the hussars (1.0 km or 1000 m) minus the distance the hussars travel towards the cliff (45.2 m):
Total Horizontal Distance = 1000 - 45.2
Total Horizontal Distance ≈ 954.8 meters
Step 4: Calculate the initial velocity required to cover the total horizontal distance in the calculated time.
The initial velocity of the cannonball can be calculated using the formula for horizontal motion:
Initial Velocity = Distance / Time
Initial Velocity ≈ 954.8 / 4.52
Initial Velocity ≈ 211.11 m/s
Therefore, the cannon shell should be fired with an initial velocity of approximately 211.11 m/s to reach the platoon of Hussars situated 1.0 km away (assuming the hussars are moving with a constant speed of 10 m/s towards the cliff).
To find the initial velocity that the cannon shell should be fired, we can use the concept of relative motion.
First, let's find the time it takes for the cannonball to reach the platoon of Hussars. Since the cannonball is fired horizontally, it will have the same horizontal velocity as the platoon, which is 10 m/s.
Let's assume the time taken for the cannonball to hit the platoon is 't'. In this time, the platoon would have moved a distance of 1 km (1000 m) towards the cliff. So, the distance covered by the platoon can be calculated using the equation:
Distance = Speed x Time
1000 m = 10 m/s x t
Simplifying the equation, we have:
t = 1000 m / 10 m/s
t = 100 s
Therefore, it will take 100 seconds for the cannonball to reach the platoon.
Now, let's calculate the vertical distance the cannonball falls during this time. The time it takes for the cannonball to reach the platoon is the same as the time it takes for the cannonball to fall vertically from the height of the cliff (100 m). We can use the equation:
Distance = (1/2) x Acceleration x Time^2
Since the cannonball is falling vertically, the acceleration due to gravity is acting only in the vertical direction. So, we can use the value '9.8 m/s^2' for acceleration and substitute the values:
100 m = (1/2) x 9.8 m/s^2 x (100 s)^2
Simplifying the equation, we have:
100 m = (1/2) x 9.8 m/s^2 x 10,000 s^2
100 m = 4,900 m/s^2 x 10,000 s^2
100 m = 49,000,000 m^2/s^2
To get rid of the square term, we take the square root of both sides:
√(100 m) = √(49,000,000 m^2/s^2)
10 m = 7,000 m/s
Therefore, the vertical component of the cannonball's velocity when it reaches the platoon is 7,000 m/s.
Since the cannonball was fired horizontally, the initial vertical velocity is 0 m/s. Therefore, the initial velocity of the cannonball can be calculated using the Pythagorean theorem:
Initial Velocity = √(Horizontal Velocity^2 + Vertical Velocity^2)
Initial Velocity = √(10 m/s)^2 + (7,000 m/s)^2)
Initial Velocity = √100 m^2/s^2 + 49,000,000 m^2/s^2
Initial Velocity = √49,000,100 m^2/s^2
Initial Velocity ≈ 7000 m/s
Hence, the initial velocity at which the cannon shell should be fired is approximately 7000 m/s.