To test out your new foam dart blaster, you hang a small cube, size 5cm on a side with mass 0.4 kg, from your ceiling. You then fire a 46 gram foam dart at the bottom of it with a speed of 15.3 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, we need to apply the principle of conservation of momentum. Here's how you can do it step by step:
Step 1: Determine the initial momentum of the foam dart.
Momentum (p) is given by the equation:
p = mass × velocity
The mass of the foam dart is 46 grams, which is equivalent to 0.046 kg. The initial velocity is 15.3 m/s.
p(dart) = 0.046 kg × 15.3 m/s
p(dart) ≈ 0.7038 kg·m/s
Step 2: Calculate the combined mass of the foam dart and the cube.
The mass of the cube is given as 0.4 kg. Adding the mass of the foam dart, the combined mass is:
Total mass = mass(dart) + mass(cube)
Total mass = 0.046 kg + 0.4 kg
Total mass = 0.446 kg
Step 3: Apply the principle of conservation of momentum.
According to the principle of conservation of momentum, the total momentum before the collision (when the foam dart hits the cube) should be equal to the total momentum after the collision.
Let's assume that the height the block reaches is h. At the topmost point of its motion, the block momentarily comes to rest, and all its initial kinetic energy is converted to potential energy.
Initial momentum = Final momentum
0 + p(dart) = 0 + (total mass × g × h)
Here, g represents the acceleration due to gravity (~9.8 m/s^2).
0.7038 kg·m/s = 0.446 kg × 9.8 m/s^2 × h
Step 4: Solve for h.
Dividing both sides of the equation by (0.446 kg × 9.8 m/s^2), we get:
h = (0.7038 kg·m/s) / (0.446 kg × 9.8 m/s^2)
h ≈ 0.14896 m
Therefore, the block will reach a height of approximately 0.14896 meters relative to where it started.