A pellet (20g) is launched horizontally off a cliff (20m high) from a spring-loaded gun. The spring has constan k=1800 N/m and is compressed 0.10m before launch.
How much kinetic energy does the pellet have just before it hits the ground?
How fast will the pellet be going just before it hits the ground?
The spring is compressed twice as much for a second firing.
How fast will the pellet be moving just after launch? How far will it go horizontally before it hits the ground?
KE=spring PE+gravitational PE
KE=1/2 k x^2+ m*g*20m
at ground level,
1/2 m vf^2=1/2 k x^2 + 20mg
solve for vf.
b. if compressed twice as much, the spring has 4x as much PE,
find the time to fall 20m
20=1/2 g t^2 solve for time t.
then, how far horizontal=20t
Do the m and 1/2 cancel each other out for the ground level question?
NO. They don't cancel. But they're given.
To find the kinetic energy of the pellet just before it hits the ground, we need to first calculate the potential energy stored in the spring and then convert it to kinetic energy.
1. Potential energy stored in the spring:
The potential energy stored in a compressed spring can be calculated using the formula:
Potential Energy (PE) = (1/2)kx²,
where k is the spring constant and x is the displacement (compression) of the spring.
Given that the spring constant (k) is 1800 N/m and the spring is compressed by 0.10m, we can calculate the potential energy stored in the spring before launch:
PE = (1/2) * 1800 * (0.10)² = 9 J (Joules)
2. Converting potential energy to kinetic energy:
The total mechanical energy of the pellet remains constant throughout its flight. The potential energy at the top of the cliff gets converted entirely into kinetic energy just before it hits the ground (assuming no energy losses due to air resistance or other factors).
Therefore, the kinetic energy of the pellet just before it hits the ground is equal to the potential energy stored in the spring:
Kinetic Energy (KE) = 9 J (Joules)
To find the speed of the pellet just before it hits the ground, we can use the formula for kinetic energy:
Kinetic Energy (KE) = (1/2)mv²,
where m is the mass of the pellet and v is its velocity (speed).
Given that the mass of the pellet is 20g (0.02kg):
9 = (1/2) * 0.02 * v²
v² = 9 / (0.02 * 0.5)
v² = 900
v ≈ 30 m/s (approximately)
Therefore, the pellet will be moving with a speed of approximately 30 m/s just before it hits the ground.
For the second firing, where the spring is compressed twice as much (0.20m), we can calculate the speed of the pellet just after launch and the horizontal distance it will travel before hitting the ground.
3. Speed of the pellet just after launch:
The potential energy stored in the spring is converted into both kinetic energy and gravitational potential energy at the instant the spring is released. We can calculate the kinetic energy just after launch by using the spring's potential energy, as we did before.
Using the formula:
Kinetic Energy (KE) = (1/2)mv²,
where m is the mass of the pellet and v is its velocity (speed).
The potential energy stored in the spring is given by:
PE = (1/2)kx²
Given that the spring constant (k) is 1800 N/m and the spring is compressed by 0.20m, we can calculate the potential energy stored in the spring just after launch:
PE = (1/2) * 1800 * (0.20)² = 36 J (Joules)
Therefore, the kinetic energy of the pellet just after launch is equal to the potential energy stored in the spring:
KE = PE = 36 J (Joules)
To calculate the speed of the pellet just after launch, we use the same formula as before:
KE = (1/2)mv²
v² = (2 * KE) / m
v² = (2 * 36) / 0.02
v² = 3600
v ≈ 60 m/s (approximately)
Therefore, the pellet will be moving with a speed of approximately 60 m/s just after launch.
4. Horizontal distance traveled before hitting the ground:
To calculate the horizontal distance traveled before hitting the ground, we need to consider the time of flight and the horizontal velocity of the pellet.
The time of flight can be calculated using the formula:
Time (t) = 2 * vertical displacement / g,
where g is the acceleration due to gravity (9.8 m/s²).
Given that the vertical displacement (height of the cliff) is 20m, we can calculate the time of flight:
t = 2 * 20 / 9.8 ≈ 4.08 seconds
The horizontal distance traveled can be calculated using the formula:
Horizontal distance (d) = horizontal velocity * time of flight
Given that the horizontal velocity is the same as the speed just after launch (60 m/s) and the time of flight is 4.08 seconds:
d = 60 * 4.08 ≈ 244.8 meters
Therefore, the pellet will travel approximately 244.8 meters horizontally before hitting the ground.