A stone is thrown straight upward with a speed of 12 m/s from the edge of a 70 m high cliff. How long is the stone in the air? What is the speed just before it hits the ground? What is the total distance the stone traveled?
(a) Height of object from cliff base
= H = 70 +12 t - (g/2) t^2
Set H = 0 and g = 9.8 m/s^2, and solve for t. Take the positive rooot.
(b) Vfinal^2 = Vinitial^2 + 2*g*70
= (144 +1372) m^2/s^2
Vfinal = 38.9 m/s
(c) (Up distance) + (Down distance)
= 2*(Up distance) + 70 m
= 2*Vo^2/(2*g)+70
= (Vo^2/g) + 70
To calculate the time the stone is in the air, we can use the equation of motion for vertical motion:
h = vit - (1/2)gt^2
Where:
h = Initial height from which the stone is thrown (70 meters)
vi = Initial velocity (12 m/s)
g = Acceleration due to gravity (-9.8 m/s^2)
t = Time
We need to solve for t, so let's rearrange the equation:
(1/2)gt^2 - vit + h = 0
Plugging in the values, we get:
(1/2)(-9.8)t^2 + 12t + 70 = 0
Solving this quadratic equation will give us the time the stone is in the air. Let's calculate it step by step:
Step 1: Multiply both sides of the equation by 2 to eliminate the fraction:
(-9.8)t^2 + 24t + 140 = 0
Step 2: Subtract 140 from both sides of the equation to isolate the t terms:
(-9.8)t^2 + 24t = -140
Step 3: Divide every term in the equation by -4.9 to simplify the equation:
t^2 - (24/(-9.8))t = -140/(-9.8)
t^2 + 2.4489t = 14.2857
Step 4: Add 2.4489/2 to both sides of the equation to complete the square:
t^2 + 2.4489t + (2.4489/2)^2 = 14.2857 + (2.4489/2)^2
t^2 + 2.4489t + 3 = 14.4556
Step 5: Simplify the equation:
(t + 1.22445)^2 = 14.4556
Step 6: Take the square root of both sides to solve for t:
t + 1.22445 = ± √14.4556
t = - 1.22445 ± √14.4556
Since time can only be positive, we take the positive root:
t = - 1.22445 + √14.4556
Using a calculator, we can find:
t ≈ 4.49 seconds
Therefore, the stone is in the air for approximately 4.49 seconds.
To calculate the speed just before it hits the ground, we use the equation:
vf = vi + gt
Where:
vf = Final velocity
vi = Initial velocity (12 m/s)
g = Acceleration due to gravity (-9.8 m/s^2)
t = Time (4.49 seconds)
Plugging in the values, we get:
vf = 12 + (-9.8)(4.49)
vf = 12 - 43.802
vf ≈ -31.802 m/s
Since the velocity is negative, it means the stone is moving downward. The magnitude of the velocity is approximately 31.802 m/s.
To calculate the total distance the stone traveled, we need to add the distance it traveled upwards and downwards. The distance traveled upwards is given by the equation:
distance_up = vi * time + (1/2) * g * time^2
distance_up = 12 * 4.49 + (1/2) * (-9.8) * (4.49)^2
Using a calculator, we can find:
distance_up ≈ 107.19 meters
The distance traveled downwards is given by the equation:
distance_down = 70 - distance_up
distance_down = 70 - 107.19
distance_down ≈ -37.19 meters
Since the distance is negative, it means the downward distance is in the opposite direction of the positive vertical axis. Therefore, we take the absolute value of the distance:
distance_down ≈ 37.19 meters
Now, we can calculate the total distance traveled by adding the upward and downward distances:
total_distance = distance_up + distance_down
total_distance = 107.19 + 37.19
total_distance ≈ 144.38 meters
Therefore, the stone traveled a total distance of approximately 144.38 meters.
To find the time the stone is in the air, we can use the following formula:
Time = (Final Velocity - Initial Velocity) / Acceleration
In this case, we can assume the acceleration due to gravity is -9.8 m/s^2 since the stone is moving upwards against gravity. The final velocity we are looking for is 0 m/s because the stone will momentarily stop at the highest point and then start falling back down.
Time = (0 m/s - 12 m/s) / -9.8 m/s^2
Time = -12 m/s / -9.8 m/s^2
Time ≈ 1.22 seconds
So, the stone is in the air for approximately 1.22 seconds.
To find the speed just before it hits the ground, we can use another formula:
Final Velocity = Initial Velocity + (Acceleration × Time)
In this case, the acceleration is -9.8 m/s^2 (negative because it's against the direction of initial velocity), and the time is the total time of flight, which we found to be 1.22 seconds. The initial velocity is the same as the final velocity when it was thrown upward, which is 12 m/s.
Final Velocity = 12 m/s + (-9.8 m/s^2 × 1.22 s)
Final Velocity ≈ -0.76 m/s
The negative sign indicates that the stone is moving downward just before hitting the ground. So, the speed just before it hits the ground is approximately 0.76 m/s.
To find the total distance the stone traveled, we can calculate the distance covered during upward motion and downward motion separately and then sum them.
Distance = (Initial Velocity × Time) + (0.5 × Acceleration × Time^2)
In the upward motion:
Distance_upward = (12 m/s × 1.22 s) + (0.5 × (-9.8 m/s^2) × (1.22 s)^2)
Distance_upward ≈ 14.73 m
In the downward motion, the stone covers the full height of the cliff, which is 70 m.
Hence, the total distance traveled by the stone is:
Distance_total = Distance_upward + Distance_downward
Distance_total ≈ 14.73 m + 70 m
Distance_total ≈ 84.73 m
So, the total distance traveled by the stone is approximately 84.73 meters.