A STUNTMAN DRIVES A MOTORCYCLE OFF A 350 m CLIFF GOING 70 mph. THE ANGLE OF ELEVATION OF THE CLIFF IS 21 HE IS HOPING TO MAKE IT ACROSS A 261 M WIDE RIVER AND LAND ON A LEDGE 82 M HIGH. DOES HE MAKE IT?
First, let's convert the given information into meters and seconds to maintain consistency of units.
1 mph = 0.44704 m/s, so 70 mph = 31.293 m/s
Now, we can break down the initial velocity into horizontal and vertical components:
Initial horizontal velocity (Vx) = initial velocity * cos(angle) = 31.293 m/s * cos(21 degrees) = 29.363 m/s
Initial vertical velocity (Vy) = initial velocity * sin(angle) = 31.293 m/s * sin(21 degrees) = 11.225 m/s
We must now find how long it takes for the stuntman to reach the ledge's height (82 m). To do this, we can use the following equation:
final vertical position = initial vertical position + (initial vertical velocity * time) - (0.5 * g * time^2)
g = 9.81 m/s^2 (acceleration due to gravity)
82 = 0 + (11.225 * time) - (0.5 * 9.81 * time^2)
Now, solve the quadratic equation for time (t):
9.81t^2 - 11.225t + 82 = 0
This gives us two solutions, however one of them will be negative and invalid for this situation.
t ≈ 3.143 s
Now that we know the time it takes to reach the ledge's height, let's find out how far horizontally the stuntman will travel in that time:
horizontal distance = initial horizontal velocity * time = 29.363 m/s * 3.143 s ≈ 92.25 m
As the distance required to cross the river is 261 m, the stuntman does not make it to the other side.
Well, if a stuntman is driving a motorcycle off a 350-meter cliff, I hope he packed some parachutes! I mean, a motorcycle isn't exactly known for its flying capabilities. But hey, let's do some calculations for fun!
The angle of elevation of the cliff is 21 degrees, which means that the vertical height of the cliff (opposite side) would be 350 meters x sin(21) ≈ 124.6 meters.
Now, the stuntman is hoping to make it across a 261-meter wide river and land on a ledge 82 meters high. We can find the distance he needs to travel horizontally as follows:
Distance = 261 meters x cos(21) ≈ 241.2 meters.
So, if the stuntman can jump off the cliff with a horizontal distance of 241.2 meters and a vertical height of 124.6 meters, he might have a shot at making it! Let's hope he sticks the landing and doesn't become a "cliff hanger"!
To determine if the stuntman makes it across the river and lands on the ledge, we can break down the problem into two parts: the horizontal distance the motorcycle travels and the vertical distance it needs to clear.
First, let's calculate the horizontal distance the motorcycle travels. We can use the formula:
Horizontal distance = Speed x Time
Given that the motorcycle is traveling at 70 mph, which is approximately 31.3 m/s, we need to calculate the time it takes for the motorcycle to reach the river. To do this, we can use the formula:
Time = Distance / Speed
Given a distance of 350 m and a speed of 31.3 m/s, we get:
Time = 350 m / 31.3 m/s = 11.16 s
Now let's calculate the horizontal distance using the time:
Horizontal distance = Speed x Time = 31.3 m/s x 11.16 s = 349.808 m
The horizontal distance is approximately 349.808 m.
Next, let's determine if the motorcycle clears the vertical distance of the ledge. The vertical distance the motorcycle needs to clear is the sum of the height of the cliff (82 m) and the height of the ledge (82 m), which gives us a total vertical distance of 164 m.
Since the stuntman is jumping off the cliff at an angle of elevation of 21 degrees, we can use trigonometry to find the height the motorcycle reaches:
Height = Horizontal distance x tan(angle of elevation)
Height = 349.808 m x tan(21 degrees) = 130.19 m
The height the motorcycle reaches is approximately 130.19 m.
Comparing the height reached (130.19 m) with the total vertical distance the motorcycle needs to clear (164 m), we can conclude that the motorcycle does not make it across the river and land on the ledge.
To determine whether the stuntman makes it across the river and lands on the ledge, we first need to find the horizontal distance he will travel before reaching the river. We can do this using trigonometry.
1. Calculate the height of the cliff, given the angle of elevation and the horizontal distance:
height of the cliff = tan(angle of elevation) * horizontal distance
height of the cliff = tan(21°) * 350m
height of the cliff ≈ 128.30m
2. Subtract the height of the cliff from the height of the ledge to find the vertical distance the stuntman needs to clear:
vertical distance to clear = height of the ledge - height of the cliff
vertical distance to clear = 82m - 128.30m
vertical distance to clear ≈ -46.30m
Since the resulting vertical distance is negative, this means that the ledge is lower than the cliff height. Therefore, the stuntman cannot make it across the river and land on the ledge.