Billy Bob “Bubba” Hickman is driving along one day messing with his cell phone instead of paying attention to what he is doing. He drives right off of a 25.0 foot high cliff at 42.4 mph.
How far from horizontally from the edge of the cliff does his truck land?
What is the direction and magnitude of Bubba’s truck’s velocity when it hits the ground?
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His horizontal speed is 42.4 mph until he stops on the ground.
I have a hunch that you are using feet and stuff so will call gravity g = 32 ft/s^2
42/4 mph = 61.6 ft/s
Now how long to fall 25 ft?
25 =(1/2) g t^2 = 16 t^2
so t^2 = 25/16
t =5/4 seconds
now how far horizontal?
x = u t = 61.6 * 5/4
================
u = 61.6
v = g t = 32 *5/4 = 40 ft/s
so speed = sqrt (40^2 + 61.6^2)
tan angle below horizontal = (40/61.6)
so tan^-1(40/61.6)
To find the distance from the edge of the cliff where Billy Bob's truck lands, we can use the horizontal (x) component of velocity and the time it takes for the truck to reach the ground.
First, let's find the time it takes for the truck to reach the ground. We'll assume there is no air resistance in this scenario, so the only force acting on the truck will be gravity, causing it to undergo free fall. The distance fallen (25.0 feet) and the acceleration due to gravity (32.2 ft/s^2) can be used to find the time of fall using the equation:
s = ut + (1/2)gt^2
Where:
s = distance fallen = 25.0 feet
u = initial vertical velocity (which is 0 in this case)
g = acceleration due to gravity = 32.2 ft/s^2
t = time of fall
Plugging in the known values and solving for t:
25.0 = 0 + (1/2)(32.2)t^2
50.0 = 32.2t^2
t^2 = 1.553
Taking the square root of both sides, we get:
t ≈ 1.246 seconds
Now, let's find the horizontal distance from the edge of the cliff:
The horizontal distance traveled (x) can be calculated using the equation:
x = horizontal velocity * time
Given that the truck's speed is 42.4 mph, we need to convert it to feet per second by multiplying it by the conversion factor:
42.4 mph * (5280 ft/1 mile) * (1 hr/3600s) ≈ 62.27 ft/s
Plugging in the value for t and the horizontal velocity:
x = 62.27 ft/s * 1.246 s ≈ 77.66 feet
Therefore, Billy Bob's truck lands approximately 77.66 feet horizontally from the edge of the cliff.
To determine the direction and magnitude of the truck's velocity when it hits the ground, we can use vector addition. The vertical and horizontal components of velocity can be combined to find the resultant velocity.
The magnitude of the resultant velocity (v) can be calculated using the Pythagorean theorem:
v = √(v_x^2 + v_y^2)
Where:
v_x = horizontal velocity
v_y = vertical velocity (which is determined by the acceleration due to gravity)
Using the values from before:
v_x = 62.27 ft/s
v_y = g * t
Plugging in the values and solving for v:
v = √(62.27^2 + (32.2 * 1.246)^2)
v ≈ √(3876.87 + 1300.70)
v ≈ √5177.57
v ≈ 71.95 ft/s
Therefore, the magnitude of Bubba's truck's velocity when it hits the ground is approximately 71.95 ft/s.
To determine the direction, we can use trigonometry. The angle (θ) can be calculated using the equation:
θ = arctan(v_y / v_x)
Plugging in the values:
θ = arctan((32.2 * 1.246) / 62.27)
θ ≈ arctan(0.5024)
θ ≈ 27.83 degrees
Therefore, the direction of Bubba's truck's velocity when it hits the ground is approximately 27.83 degrees above the horizontal.