A cannon that is 10m long is designed to launch a 10kg ball over a castle wall.
In order to do this the ball must have a speed of at least 50m/s as it exits the
cannon. For every 10kg of explosives used, the force on the ball in the cannon
increases by 1000N. How many kg of explosives should they use?
2 Gravitation(100pts.)
To calculate the number of kilograms of explosives required, we need to determine the force needed to launch the ball at a speed of 50 m/s and then determine the increase in force for every 10 kg of explosives used.
First, let's determine the force required to launch the ball. We can use the principle of conservation of energy, which states that the potential energy of the ball at the top of its trajectory is equal to the kinetic energy of the ball just as it leaves the cannon.
The potential energy is given by the formula: PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the castle wall. Since no height is given, we can assume the ball is launched horizontally. In this case, the height remains constant and cancels out in the calculation.
Therefore, the potential energy is zero, and the kinetic energy of the ball just as it leaves the cannon is equal to its initial energy. This is given by the formula: KE = (1/2)mv^2, where m is the mass of the ball and v is its speed.
We can rearrange the formula to solve for the mass of the ball:
m = 2 * KE / v^2
Since the mass is given as 10 kg and the speed is given as 50 m/s, we can substitute these values into the formula:
m = 2 * (10 kg) / (50 m/s)^2
m = 2 * 10 kg / 2500 m^2/s^2
m = 0.008 kg
Now that we know the mass of the ball, we can determine the force needed to accelerate it to a speed of 50 m/s. We can use Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a). In this case, acceleration is the change in velocity per unit time.
The force required is given by the formula: F = ma
Since the acceleration is the change in velocity per unit time, we can calculate it using the formula: a = (vf - vi) / t, where vf is the final velocity (50 m/s), vi is the initial velocity (0 m/s), and t is the time taken to reach the final velocity (which we assume to be 1 second as it's not specified).
Substituting the values into the formula:
a = (50 m/s - 0 m/s) / 1 s
a = 50 m/s^2
Now, substituting the values into Newton's second law:
F = (0.008 kg) * (50 m/s^2)
F = 0.4 kg m/s^2 = 0.4 N
Next, we need to determine the increase in force for every 10 kg of explosives used. According to the given information, the force increases by 1000 N for every 10 kg of explosives.
Therefore, the increase in force per unit explosives is: 1000 N / 10 kg = 100 N/kg
To find out how many kilograms of explosives should be used to achieve the required force, we can divide the required force by the increase in force per unit explosives:
Number of kilograms of explosives = Required force / Increase in force per unit explosives
Number of kilograms of explosives = 0.4 N / (100 N/kg)
Simplifying:
Number of kilograms of explosives = 0.004 kg
Therefore, they should use 0.004 kg (or 4 grams) of explosives to achieve the required force.
I hope this explanation helps! Let me know if you have any further questions.