Two objects were projected vertically upwards
at different time with a speed of 80ms-¹ and
100ms-¹ respectively.if the time interval is 2
sec,when and where would they meet.?
Step plz
same height?
h1=80*t-4.9t^2
h2=100(t-2)-4.9 (t-2)^2
set h1=h2
solve for t. Notice it is a quadratic.
with t, now solve for either h.
Step 1: Convert the speeds of the objects to meters per second (m/s):
The first object was projected upwards with a speed of 80 ms^-1
The second object was projected upwards with a speed of 100 ms^-1
Step 2: Determine the time it would take for each object to reach the highest point:
The time it takes for an object to reach the highest point and then come back down is equal to the time it takes for the object to reach its highest point divided by two.
For the first object:
Time to reach the highest point = (80 ms^-1) / 9.8 m/s² (acceleration due to gravity)
Time to reach the highest point = 8.16 seconds
Time to reach the highest point and come back down = (8.16 seconds) / 2 = 4.08 seconds
For the second object:
Time to reach the highest point = (100 ms^-1) / 9.8 m/s²
Time to reach the highest point = 10.20 seconds
Time to reach the highest point and come back down = (10.20 seconds) / 2 = 5.10 seconds
Step 3: Determine the time interval during which the objects are in the air at the same time:
The time interval during which the objects are in the air at the same time is equal to the difference in their time to reach the highest point and come back down.
Time interval = (5.10 seconds) - (4.08 seconds)
Time interval = 1.02 seconds
Step 4: Determine the meeting time and location:
Since the time interval is 2 seconds, and the meeting time occurs within this interval, we can conclude that the objects meet during their ascent.
The time it takes for the objects to meet = 2.00 seconds - 1.02 seconds
The time it takes for the objects to meet = 0.98 seconds
To find the height at which the objects meet, we can use the equation for the height of an object in free fall as a function of time:
Height = (initial velocity * time) - (0.5 * acceleration due to gravity * time^2)
For the first object:
Height = (80 ms^-1) * (0.98 seconds) - (0.5 * 9.8 m/s²) * (0.98 seconds)^2
Height ≈ 74.832 meters
For the second object:
Height = (100 ms^-1) * (0.98 seconds) - (0.5 * 9.8 m/s²) * (0.98 seconds)^2
Height ≈ 95.124 meters
Therefore, the two objects meet approximately 0.98 seconds after the second object is projected, at a height of 74.832 meters above the ground.
To find out when and where the two objects would meet, we need to analyze their vertical motion. Let's break down the steps to solve this problem:
Step 1: Determine the initial velocities of the two objects:
- One object has an initial velocity of 80 m/s upwards.
- The other object has an initial velocity of 100 m/s upwards.
Step 2: Understand the motion of each object:
- When an object is projected upwards, it experiences free fall due to gravity, which causes it to decelerate until it reaches its maximum height and starts descending.
- We need to determine how long it takes for each object to reach its maximum height and when they start descending.
Step 3: Calculate the time of flight for each object:
- The time it takes for an object to reach its maximum height can be calculated using the formula: time = (final velocity - initial velocity) / acceleration.
- In this case, the acceleration is due to gravity, which is approximately -9.8 m/s².
For the object with an initial velocity of 80 m/s:
- Initial velocity (u) = 80 m/s (upwards)
- Final velocity (v) = 0 m/s (at maximum height)
- Acceleration (a) = -9.8 m/s² (downwards)
Using the formula: time = (v - u) / a
=> time = (0 - 80) / -9.8
=> time ≈ 8.16 seconds
For the object with an initial velocity of 100 m/s:
- Initial velocity (u) = 100 m/s (upwards)
- Final velocity (v) = 0 m/s (at maximum height)
- Acceleration (a) = -9.8 m/s² (downwards)
Using the formula: time = (v - u) / a
=> time = (0 - 100) / -9.8
=> time ≈ 10.20 seconds
Step 4: Determine the meeting time:
- As per the problem, the time interval between the two objects is 2 seconds.
- We need to determine when the two objects will meet.
Meeting time = time of flight of the object with an initial velocity of 80 m/s + time interval
≈ 8.16 seconds + 2 seconds
≈ 10.16 seconds
Step 5: Calculate the meeting position:
- Since both objects will meet at the same time, we need to find the height they will be at that time.
Meeting position = (initial velocity of the object with an initial velocity of 80 m/s) * (meeting time)
≈ 80 m/s * 10.16 seconds
≈ 812.8 meters
Therefore, the two objects would meet approximately 10.16 seconds after the lower object was projected, and the meeting point would be at a height of approximately 812.8 meters.