A 5.20 g pellet is shot horizontally from a BB gun at a speed of 29.1 m/s into a 20.8 g wooden block. The wooden block is attached to a spring and lies on a frictionless table. If the collision is inelastic and the spring constant k = 22.0 N/m, what is the maximum compression of the spring?

Question

A 5.20 g pellet is shot horizontally from a BB gun at a speed of 29.1 m/s into a 20.8 g wooden block. The wooden block is attached to a spring and lies on a frictionless table. If the collision is inelastic and the spring constant k = 22.0 N/m, what is the maximum compression of the spring?

Answer 1

To find the maximum compression of the spring, we first need to calculate the initial momentum of the pellet and the block, and then use the conservation of momentum to find their final velocity. From there, we can determine the maximum compression of the spring using Hooke's Law.

Step 1: Calculate the initial momentum of the pellet and the block.
Momentum (p) is calculated using the formula: p = mass × velocity.
For the pellet: mass = 5.20 g = 0.00520 kg.
velocity = 29.1 m/s.
So, the initial momentum of the pellet (p1) is:
p1 = (0.00520 kg) × (29.1 m/s)

For the block: mass = 20.8 g = 0.0208 kg.
Since the block is at rest initially, its initial momentum (p2) is zero.

Step 2: Apply the conservation of momentum.
In an inelastic collision, momentum is conserved. So, the total momentum before the collision (p1 + p2) is equal to the total momentum after the collision.

Step 3: Calculate the final velocity of the pellet and the block.
Let's assume the final velocity of the combined pellet and block system is Vf.
Using the conservation of momentum, we have:
p1 + p2 = (mass of pellet + mass of block) × Vf.

Step 4: Calculate the maximum compression of the spring using Hooke's Law.
When the pellet and the block collide, their kinetic energy is converted into the potential energy stored in the spring.
The potential energy stored in a spring is given by the equation: PE = (1/2)kx^2.
Where PE is the potential energy, k is the spring constant, and x is the compression (displacement) of the spring.

At the maximum compression of the spring, the kinetic energy of the system becomes zero because all the energy is stored as potential energy in the spring.

So, we can equate the initial kinetic energy to the final potential energy:
(1/2)(mass of pellet + mass of block)Vf^2 = (1/2)kx^2.

Rearranging the equation, we have:
x^2 = (mass of pellet + mass of block)Vf^2 / k.

Taking the square root of both sides, we get:
x = √((mass of pellet + mass of block)Vf^2 / k).

Finally, plug in the values into the equation and calculate the maximum compression (x) of the spring.