You shoot a 57 gram dart vertically upwards from a catapult with a spring constant of 379 N/m. The catapult is initially stretched from the equilibrium point by 30 cm. What is the height above the starting point reached by the dart? Ignore air resistance and give your answer to 2 s.f.
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To find the height reached by the dart, we need to use the conservation of mechanical energy.
The potential energy stored in the spring when it is stretched by 30 cm is given by:
Potential energy (PE) = 0.5 * k * x^2,
where k is the spring constant and x is the displacement from the equilibrium point.
Using the given values, we have:
PE = 0.5 * 379 N/m * (0.3 m)^2
= 0.5 * 379 N/m * 0.09 m^2
= 17.055 N*m
This potential energy gets converted into gravitational potential energy as the dart reaches its maximum height.
The gravitational potential energy (PE) is given by:
PE = m * g * h,
where m is the mass of the dart, g is the acceleration due to gravity, and h is the height reached by the dart.
Rearranging the equation, we have:
h = PE / (m * g).
Substituting the given values, we have:
h = 17.055 N*m / (0.057 kg * 9.8 m/s^2)
= 30.789 m.
Therefore, the height above the starting point reached by the dart is approximately 30.79 meters (to 2 significant figures).
To find the height reached by the dart, we need to consider the potential energy stored in the spring of the catapult being converted into gravitational potential energy of the dart.
First, let's find the potential energy stored in the spring of the catapult. The potential energy (PE) stored in a spring is given by the formula:
PE = (1/2) k x^2
where k is the spring constant and x is the displacement from the equilibrium position.
In this case, the spring constant (k) is given as 379 N/m and the displacement (x) is 30 cm, which is equal to 0.30 m.
PE = (1/2) * 379 * (0.30)^2
= 20.7655 J (to 4 decimal places)
Now, let's consider the potential energy being converted into the gravitational potential energy of the dart. The gravitational potential energy (PE_gravity) is given by the formula:
PE_gravity = m * g * h
where m is the mass of the dart, g is the acceleration due to gravity, and h is the height above the starting point reached by the dart.
In this case, the mass of the dart (m) is given as 57 grams, which is equal to 0.057 kg. The acceleration due to gravity (g) is approximately 9.8 m/s^2.
PE_gravity = 0.057 * 9.8 * h
= 0.5586 * h (to 4 decimal places)
Since the potential energy stored in the spring is converted into gravitational potential energy, we can equate the two expressions for potential energy:
20.7655 J = 0.5586 * h
To find h, we can rearrange the equation:
h = 20.7655 J / 0.5586
h ≈ 37.18 m (to 2 significant figures)
Therefore, the height above the starting point reached by the dart is approximately 37.18 meters.