To test out your new foam dart blaster, you hang a small cube, size 5cm on a side with mass 0.5 kg, from your ceiling. You then fire a 41.9 gram foam dart at the bottom of it with a speed of 18 m/s, and the dart sticks to the cube. Calculate the height in m that the block will reach relative to where it started.
To calculate the height the block will reach relative to where it started, we can use the principles of conservation of momentum and conservation of energy.
Let's denote the initial height of the block from the starting position as h, and the final height as H.
First, let's calculate the initial momentum of the dart:
Momentum (p) = mass (m) x velocity (v)
Momentum = 0.0419 kg x 18 m/s
Momentum = 0.7542 kg·m/s
According to the conservation of momentum, the momentum of the dart-block system remains constant after the collision.
Next, let's calculate the final velocity of the block-dart system after the collision:
Momentum (p) = mass (m) x velocity (v)
Velocity = Momentum / mass of the combined system
Velocity = (0.7542 kg·m/s) / (0.5 kg + 0.0419 kg)
Velocity = 1.4289 m/s
Now, let's use the principles of conservation of energy to calculate the height the combined block-dart system will reach.
The total initial energy is the sum of the kinetic energy (KE) and potential energy (PE) at the start:
Initial energy = KE + PE
Initial energy = 0.5 * (0.5 kg) * (0 m/s)^2 + (0.5 kg) * 9.8 m/s^2 * h
Initial energy = 0 + 4.9 h
The total final energy is the sum of the kinetic energy and potential energy when the block reaches height H:
Final energy = KE + PE
Final energy = 0.5 * (0.5 kg + 0.0419 kg) * (1.4289 m/s)^2 + (0.5 kg + 0.0419 kg) * 9.8 m/s^2 * H
Final energy = 0.5 * 0.5419 kg * (1.4289 m/s)^2 + 0.5419 kg * 9.8 m/s^2 * H
Final energy = 0.369 J + 5.3404 H
Since energy is conserved, the initial energy equals the final energy:
0.369 J + 5.3404 H = 4.9 h
Solving for H, we get:
H = (4.9 h - 0.369) / 5.3404
Now we can plug in the given values to find the value of H:
H = (4.9 * h - 0.369) / 5.3404
Assuming the starting height (h) is zero, we get:
H = (4.9 * 0 - 0.369) / 5.3404
H = -0.369 / 5.3404
H ≈ -0.069 m
Therefore, using this calculation, the block will reach approximately -0.069 meters relative to where it started.
To calculate the height the cube will reach relative to where it started, we need to consider the law of conservation of momentum.
1. Calculate the momentum of the foam dart before it is fired:
- Mass of the dart = 41.9 grams = 0.0419 kg
- Initial velocity of the dart = 18 m/s
- Momentum = mass × velocity = 0.0419 kg × 18 m/s = 0.7542 kg·m/s
2. Since the dart sticks to the cube, the momentum of the system (cube + dart) after the collision will be equal to the initial momentum of the dart.
3. Calculate the velocity of the cube and dart system after the collision:
- Mass of the cube = 0.5 kg
- Momentum after the collision = 0.7542 kg·m/s (conservation of momentum)
- Velocity after the collision = Momentum after the collision / Mass of the cube
= 0.7542 kg·m/s / 0.5 kg
= 1.5084 m/s
4. Calculate the change in kinetic energy of the system after the collision:
- Initial kinetic energy = 0.5 × mass of the cube × (initial velocity of the cube)^2
= 0.5 × 0.5 kg × 0 m/s
= 0 J (since the cube was initially at rest)
- Final kinetic energy = 0.5 × mass of the cube × (final velocity of the cube)^2
= 0.5 × 0.5 kg × (1.5084 m/s)^2
≈ 0.5678 J
5. The change in kinetic energy is equal to the work done on the system. In this case, the work done is equal to the potential energy gained by the system due to the height it reaches.
6. Calculate the height reached by the cube:
- Potential energy gained by the system = Change in kinetic energy
= 0.5678 J
- Height = Potential energy gained by the system / (mass of the cube × g)
= 0.5678 J / (0.5 kg × 9.8 m/s^2)
≈ 0.1159 m
Therefore, the cube will reach a height of approximately 0.1159 meters relative to where it started.
PE gained = KE on impact.
(0.5 + 0.0419)gh = 1/2 * 0.0419 * 18^2
solve for h