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.

Question

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.

Answer 1

To find the height above the starting point reached by the dart, we can use the principles of conservation of mechanical energy.

The potential energy stored in the stretched catapult is given by the formula:

Potential energy = (1/2)kx^2

Where:
k = spring constant (379 N/m)
x = displacement from the equilibrium point (0.30 m)

So, the potential energy stored in the catapult is:

Potential energy = (1/2)(379 N/m)(0.30 m)^2
= 17.07 J

In the absence of air resistance, the potential energy is converted into kinetic energy as the dart travels upwards. At the maximum height, all of the potential energy is converted into kinetic energy. The kinetic energy is given by the formula:

Kinetic energy = (1/2)mv^2

Where:
m = mass of the dart (57 g = 0.057 kg)
v = velocity of the dart at the maximum height (unknown)

Setting the potential energy equal to the kinetic energy, we have:

Potential energy = Kinetic energy
17.07 J = (1/2)(0.057 kg)v^2

Simplifying and solving for v^2:

v^2 = (2*17.07 J) / 0.057 kg
v^2 = 5980.7 m^2/s^2

To find the velocity, we take the square root of both sides:

v = √(5980.7 m^2/s^2)
v ≈ 77.36 m/s

Now, we can use the kinematic equation to find the height reached, since we know the initial velocity (v0 = 77.36 m/s) and the acceleration due to gravity (g ≈ 9.8 m/s^2).

The kinematic equation for vertical motion is:

v^2 = v0^2 + 2gh

Where:
v = final velocity (0 m/s at maximum height)
v0 = initial velocity (77.36 m/s)
g = acceleration due to gravity (9.8 m/s^2)
h = height above the starting point (unknown)

Rearranging the equation and solving for h:

0 = (77.36 m/s)^2 + 2(9.8 m/s^2)h
0 = 5980.7 m^2/s^2 + 19.6 m/s^2 h

Solving for h, we have:

h = -5980.7 m^2/s^2 ÷ (19.6 m/s^2)
h ≈ -305.66 m

Since the height above the starting point cannot be negative, we ignore the negative sign. Therefore, the height reached by the dart is approximately 305.66 meters.

Rounding to 2 significant figures, the answer is approximately 310 meters.