You stumble and accidentally toss your car keys horizontally at 50 m/s from a cliff that is 45 m high. How far from the base of the cliff should you look for your keys?
how long does it take to fall 45m?
4.9t^2 = 45
now, distance = speed * time
To determine how far from the base of the cliff you should look for your keys, we can use the equations of motion.
First, let's find the time it takes for the keys to fall to the ground. We can use the equation:
y = vi*t + (1/2)*a*t^2
where y is the vertical displacement (45 m), vi is the initial vertical velocity (0 m/s), a is the acceleration due to gravity (-9.8 m/s^2), and t is the time.
Rearranging the equation:
45 = (1/2)*(-9.8)*t^2
Solving for t:
t^2 = (2*45)/9.8
t^2 ≈ 9.18
t ≈ √9.18
t ≈ 3.03 s
So it takes approximately 3.03 seconds for the keys to fall to the ground.
Now let's find how far horizontally the keys will travel during this time. Since the initial horizontal velocity is 50 m/s and there are no horizontal forces acting on the keys, the horizontal displacement is given by:
x = vi*t
where x is the horizontal displacement and vi is the initial horizontal velocity.
x = 50 * 3.03
x ≈ 151.5 m
Therefore, you should look approximately 151.5 meters horizontally from the base of the cliff to find your keys.
To find out how far from the base of the cliff you should look for your keys, we can use the principles of projectile motion.
In this scenario, the horizontal velocity of the keys remains constant at 50 m/s throughout their flight, and the only force acting on them is gravity pulling them vertically downwards.
We can break down the problem into two parts: the horizontal motion and the vertical motion of the keys.
First, let's calculate the time it takes for the keys to hit the ground using the vertical motion. We can use the formula:
h = (1/2) * g * t^2
Where:
- h is the vertical distance traveled (45 m in this case)
- g is the acceleration due to gravity (approximately 9.8 m/s^2, assuming negligible air resistance)
- t is the time it takes for the keys to hit the ground
Rearranging the formula, we get:
t^2 = (2 * h) / g
Plugging in the values, we have:
t^2 = (2 * 45) / 9.8
t^2 = 9.18
t ≈ √9.18
t ≈ 3.03 seconds
Now that we know it takes approximately 3.03 seconds for the keys to hit the ground, we can calculate the horizontal distance they travel using the horizontal motion.
The horizontal distance is given by the formula:
d = v * t
Where:
- d is the horizontal distance traveled by the keys
- v is the horizontal velocity (50 m/s in this case)
- t is the time taken (3.03 seconds)
Plugging in the values, we get:
d = 50 m/s * 3.03 s
d ≈ 151.5 meters
Therefore, you should look for your keys approximately 151.5 meters from the base of the cliff.