A tennis ball is released at 30 degrees. It reaches max height of 20 ft find initial speed
So we are to assume its initial velocity was 30 deg above the horizon.
verticalspeed=vsin30
The KE in the horizontal direction remains constant, but in the vertical
1/2 m v'^2+mgh=1/2 m (V sin30)^2
at the top, v' is zero, so
g*20=1/2 V^2 *1/4)
v^2=8g*20ft=160ft*32ft/s^2
v=sqrt (160*32) ft/sec=21.6ft/s
check that.
Y^2 = Yo^2 + 2g*h = 0.
Yo^2 - 64*20 = 0, Yo = 35.8 Ft/s.
Vo*sin30 = 35.8, Vo = 71.6 Ft/s.
To find the initial speed of the tennis ball, we can use the equations of projectile motion.
When the ball is at its maximum height, its vertical velocity becomes zero. Using this information, we can proceed with the following steps:
Step 1: Convert the height from feet to meters. Since the formula requires the units to be in meters, we need to convert 20 ft to meters. 1 ft is equal to 0.3048 meters, so:
20 ft * 0.3048 meters/ft = 6.096 meters (rounded to three decimal places)
Step 2: Determine the acceleration due to gravity. Close to the surface of the Earth, the acceleration due to gravity is approximately 9.8 m/s^2 (assuming no air resistance).
Step 3: Calculate the vertical displacement, Δy, using the formula:
Δy = v₀y² / (2 * g)
Where:
Δy = vertical displacement (from the ground to the maximum height)
v₀y = initial vertical velocity (unknown)
g = acceleration due to gravity
Plugging in the values:
6.096 meters = v₀y² / (2 * 9.8 m/s^2)
Step 4: Solve for v₀y:
Multiply both sides by (2 * 9.8 m/s^2):
2 * 9.8 m/s^2 * 6.096 meters = v₀y²
v₀y² = 119.36608 m^2/s^2
Take the square root of both sides:
v₀y ≈ 10.934 m/s (rounded to three decimal places)
The initial vertical velocity is approximately 10.934 m/s.
Step 5: Find the initial speed. Since the initial velocity can be divided into its horizontal and vertical components (vx and vy), we can use trigonometry to calculate it using the angle.
We know that the initial angle of projection is 30 degrees. The initial speed can be calculated using the equation:
v₀ = v₀x / cos(θ)
Where:
v₀ = initial speed
v₀x = initial horizontal velocity
θ = angle of projection
In this case, we want to find v₀x, which is the horizontal component of the initial velocity. Therefore:
v₀x = v₀ * cos(θ)
Plugging in the values:
v₀x = 10.934 m/s * cos(30°)
v₀x ≈ 9.499 m/s (rounded to three decimal places)
So, the initial speed of the tennis ball is approximately 9.499 m/s.