A spring with k=42N/m is mounted horizontally at the edge of a 1.2m high table. The spring is compressed 5cm and a 25g pellet is placed at its end. When the spring is released,m how far horizontally from the edge of the table does the pellet strike the floor?
To find the horizontal distance the pellet strikes the floor, we need to analyze the potential energy and kinetic energy of the system.
Let's break the problem into two main parts:
1. Calculating the potential energy stored in the compressed spring.
2. Converting the potential energy into kinetic energy and calculating the horizontal distance traveled by the pellet.
Step 1: Calculating the potential energy of the compressed spring.
The potential energy (PE) stored in a spring is given by the formula: PE = (1/2) * k * x^2
Where:
- PE is the potential energy stored in the spring
- k is the spring constant (k = 42 N/m)
- x is the compression or stretching distance (x = 5 cm = 0.05 m)
Using the given values, we can calculate the potential energy stored in the compressed spring:
PE = (1/2) * 42 * (0.05)^2 = 0.0525 J
Step 2: Converting potential energy to kinetic energy and calculating the horizontal distance.
The potential energy is converted into kinetic energy when the pellet is released. The total mechanical energy (TE) remains constant throughout the system.
TE = PE + KE
TE = KE
Where:
- KE is the kinetic energy
- TE is the total mechanical energy
Since the pellet is not at an initial height, the potential energy initially stored in the spring is converted entirely into kinetic energy.
Therefore, KE = PE = 0.0525 J
The kinetic energy (KE) is given by the formula:
KE = (1/2) * m * v^2
Where:
- m is the mass of the pellet (m = 25 g = 0.025 kg)
- v is the velocity of the pellet
We have the kinetic energy (KE) and mass (m), so we can rearrange the formula to find the velocity (v):
v = sqrt((2 * KE) / m)
v = sqrt((2 * 0.0525) / 0.025)
v = sqrt(4.2) ≈ 2.05 m/s
To find the horizontal distance, we need to determine the time taken by the pellet to reach the floor from a horizontal initial velocity of 2.05 m/s. As we know initial vertical velocity (vu = 0 m/s), and the distance traveled in the vertical direction (h = 1.2 m), we can use the equation of motion:
h = vit + (1/2)gt^2
Where:
- h is the height of the table (h = 1.2 m)
- vi is the initial vertical velocity (vi = 0 m/s)
- g is the acceleration due to gravity (g ≈ 9.8 m/s^2)
- t is the time taken to reach the ground
Rearranging the equation to solve for time (t):
t = sqrt((2 * h) / g)
t = sqrt((2 * 1.2) / 9.8)
t ≈ 0.49 s
Finally, we can calculate the horizontal distance (d) traveled by the pellet:
d = v * t
d ≈ 2.05 * 0.49
d ≈ 1.00 m
Therefore, the pellet strikes the floor approximately 1.00 meter horizontally from the edge of the table.