A "Stomp Rocket" is a toy projectile launcher illustrated in figure (i) below, which uses a blast of air to propel a dart made of plastic. The air blast is produced by jumping on a plastic bladder as shown in figure (ii) below. It is found that when a 187 lb dad jumps on the bladder, the rocket dart will fly on average a horizontal distance of 470 ft when launched at an angle of 45°.
a).What is the initial velocity of the rocket dart as it leaves the launch tube?
b) If the rocket dart is instead pointed straight up, and the same dad jumps on the launcher, what would be the maximum height obtained?
There is more information here than you require to answer the question.
A projectile with initial velocity V that is lauched at an angle A to the horizontal will travel a horizontal distance
X = (2 V^2/g)sin A cos A
= (V^2/g)sin (2A)
Since A = 45 degrees, sin (2A) = 1 and
470 ft = V^2/32.2 ft/s^2
V = 123 ft/s
Use that velocity to calculate how high it will go if launched vertically
gH = V^2/2 (from conservation of energy)
H = V^2/(2g) = X/2 = 235 ft
It can go half as high as it can go horizontally.
To find the initial velocity of the rocket dart as it leaves the launch tube, we can use the range formula for projectile motion. The range is the horizontal distance traveled by the projectile.
a) The range formula for projectile motion is:
R = (V₀² * sin(2θ)) / g
Where:
R = range (470 ft in this case)
V₀ = initial velocity
θ = launch angle (45° in this case)
g = acceleration due to gravity (32.174 ft/s²)
To solve for V₀, we rearrange the formula:
V₀ = √((R * g) / sin(2θ))
Substituting the given values:
V₀ = √((470 * 32.174) / sin(90°)) ≈ √(15100 / 1) ≈ √15100 ≈ 122.82 ft/s
Therefore, the initial velocity of the rocket dart as it leaves the launch tube is approximately 122.82 ft/s.
b) To find the maximum height when the rocket dart is launched straight up, we need to use the maximum height formula for projectile motion. The maximum height is the highest point reached by the projectile.
The maximum height formula is:
H = (V₀² * sin²(θ)) / (2g)
Where:
H = maximum height
V₀ = initial velocity (calculated in part a)
θ = launch angle (90° for straight up)
g = acceleration due to gravity (32.174 ft/s²)
Substituting the given values:
H = (122.82² * sin²(90°)) / (2 * 32.174) ≈ (15100 * 1) / 64.348 ≈ 235 ft
Therefore, the maximum height obtained when the rocket dart is launched straight up would be approximately 235 ft.