A catapult is used to project a stone of mass 15g vertically upwards. The rubber used in the catapult has a constant force of 30NM–1. If the rubber is stretched by a force of 30N, calculate the velocity at which the stone was projected
PE = 1/2 kx^2 = 1/2 mv^2
you have m,k,x, so find v.
Well, let's see. We have a catapult, a stone, and some rubber. It's like a circus act in the making!
Now, we know that the rubber has a constant force of 30 Newtons per meter. And if we stretch the rubber by a force of 30 Newtons, we can calculate the distance it stretches.
Using Hooke's Law, which relates the force applied to an elastic material and the distance it stretches, we can say:
Force = k * distance
Here, the force is 30 Newtons, and k is the constant force of 30 Newtons per meter. So, if we solve for the distance, we get:
30 N = 30 N/m * distance
Dividing both sides by 30 N/m, we find that the distance is 1 meter.
Now, to find the velocity at which the stone was projected, we can use the formula for potential energy:
Potential Energy = 0.5 * m * v^2
Where m is the mass of the stone, and v is its velocity. We can rearrange this equation to solve for v:
v^2 = (2 * Potential Energy) / m
Given that the mass of the stone is 15 grams (0.015 kilograms), and the potential energy is the force times the distance (30 N * 1 m = 30 J), we can substitute these values into the equation:
v^2 = (2 * 30 J) / 0.015 kg
v^2 = 4000 m^2/s^2
Taking the square root of both sides, we find that the velocity at which the stone was projected is approximately 63.25 m/s.
So, watch out for flying stones and clowns when launching that catapult!
To calculate the velocity at which the stone is projected, we can use the principle of conservation of energy. Since the stone is being projected vertically upwards, only the potential energy and kinetic energy will come into play.
First, let's find the potential energy of the stone when it is at the highest point (considered as zero height). The potential energy (PE) is given by the formula:
PE = m * g * h
Where:
m = mass of the stone in kilograms
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height of the stone at the highest point
In this case, the mass of the stone is 15g, but we need to convert it to kilograms:
m = 15g / 1000 (since 1kg = 1000g)
m = 0.015 kg
Since the stone is projected vertically upwards, its height at the highest point is equal to the maximum stretch of the rubber in the catapult, which is 30N. Therefore:
h = 30N / 30N/m (using the rubber's constant force)
h = 1 meter
Now we can calculate the potential energy:
PE = 0.015 kg * 9.8 m/s^2 * 1 m
PE = 0.147 J
Next, let's find the kinetic energy (KE) of the stone when it is at the highest point. The kinetic energy is given by the formula:
KE = (1/2) * m * v^2
Where:
m = mass of the stone in kilograms
v = velocity of the stone when projected
Since we want to find the velocity, we rearrange the formula to solve for v:
v = sqrt((2 * KE) / m)
In this case, the potential energy at the highest point represents the maximum kinetic energy the stone will have. Therefore:
KE = 0.147 J
Now we can calculate the velocity:
v = sqrt((2 * 0.147 J) / 0.015 kg)
v = sqrt(19.6 m^2/s^2)
v ≈ 4.43 m/s
Therefore, the stone was projected vertically upwards with a velocity of approximately 4.43 m/s.