during a scienceday learners use a plastic cold drink bottle as a rocket. they use a bicycle pump to increase the bottle. when the pressure becomes high enough, the bottle is launched vertically upward into the air. the bottle returns to its original position after 3s. ignore the effects of air friction
To calculate the maximum height reached by the rocket, we can use the principles of projectile motion.
Step 1: We need to determine the initial velocity of the rocket. The initial velocity is essentially the speed at which the rocket leaves the ground when launched.
Step 2: Since the rocket returns to its original position after 3 seconds, we know that the total time of flight is 6 seconds (2 times the time taken to reach the maximum height).
Step 3: We can use the formula for vertical motion to calculate the maximum height (h) reached by the rocket:
h = v₀ * t + (1/2) * g * t²
where:
- h is the maximum height
- v₀ is the initial vertical velocity (positive upwards)
- t is the time of flight
- g is the acceleration due to gravity (approximately 9.8 m/s²)
Step 4: Since the rocket starts and ends at the same height, the initial and final heights are the same. Therefore, the initial vertical velocity of the rocket (v₀) is equal to 0.
Step 5: Plug in the values into the formula:
h = (0) * (3) + (1/2) * (9.8) * (3)^2
Step 6: Simplify the equation:
h = 0 + (1/2) * (9.8) * 9
h = 0 + 4.9 * 9
h = 44.1 meters
Therefore, the maximum height reached by the rocket is 44.1 meters.
To calculate the maximum height reached by the bottle, we can use the principles of projectile motion. Assuming there are no external forces acting on the bottle after it is launched, we can break down the motion into two components: vertical and horizontal.
First, let's calculate the time it takes for the bottle to reach its maximum height. We are told that the bottle returns to its original position after 3 seconds, which means it spends half of that time going up and half going down. Therefore, the time to reach the maximum height is 3s/2 = 1.5 seconds.
Now, let's calculate the maximum height using the formula for vertical motion:
h = v₀t - 0.5gt²
Here,
h = maximum height (what we want to find)
v₀ = initial vertical velocity (when the bottle is launched)
t = time to reach the maximum height (1.5 seconds)
g = acceleration due to gravity (approximately 9.8 m/s²)
Since the bottle is launched vertically upward, its initial vertical velocity is positive. However, we don't know the exact value of v₀, so we need to determine it through further calculation.
To determine v₀, we can use the formula for speed of an object, assuming no air friction:
v = u + at
Here,
v = final vertical velocity (0 m/s, as the bottle comes to a stop at the maximum height)
u = initial vertical velocity (what we want to find)
a = acceleration due to gravity (-9.8 m/s², since gravity opposes the upward motion)
Rearranging the equation, we get:
u = v - at
Substituting the values:
u = 0 - (-9.8 × 1.5)
Simplifying,
u = 0 + 14.7
u = 14.7 m/s
Now that we know the initial vertical velocity, we can calculate the maximum height by substituting the values into the first equation:
h = v₀t - 0.5gt²
h = 14.7 × 1.5 - 0.5 × 9.8 × (1.5)²
Simplifying,
h = 22.05 - 0.5 × 9.8 × 2.25
h = 22.05 - 10.20675
h ≈ 11.84325 m
Therefore, the maximum height reached by the bottle is approximately 11.84 meters.