Sam while sitting in a flat car at rest notices a plum about to fall from a tall plumb tree. The height of the plum from over the Sam is 19.6 m, and Sam is 29.4 m to get directly under the falling plum. If Sam must catch the plum just before it hits the ground calculate (a) the acceleration of his flat car to get him there quickly, (b) the time it would requires
To calculate the acceleration of the flat car, we can use the equation of motion for an object in free fall:
h = (1/2)gt^2,
where h is the height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time. Since the plum is at a height of 19.6 m and Sam is 29.4 m away from the point directly under the plum, the total distance Sam needs to close is 19.6 + 29.4 = 49 m.
Using the equation of motion, we can rearrange it to solve for t:
t = sqrt(2h/g).
Plugging in the values, we get:
t = sqrt(2 * 19.6 / 9.8) = sqrt(4) = 2 s.
Therefore, it will take Sam 2 seconds to get directly under the falling plum.
To find the acceleration of the flat car, we can use the formula:
a = (vf - vi) / t,
where vf is the final velocity, vi is the initial velocity (which is 0 since the car is at rest), and t is the time. We know that vf is the distance traveled divided by the time:
vf = (49 m) / (2 s) = 24.5 m/s.
Now we can calculate the acceleration:
a = (24.5 m/s - 0 m/s) / (2 s) = 12.25 m/s^2.
Therefore, Sam needs his flat car to accelerate at a rate of 12.25 m/s^2 in order to catch the plum just before it hits the ground.
To calculate the acceleration of Sam's flat car and the time it would require for Sam to catch the falling plum, we can use the following steps:
Step 1: Determine the time it takes for the plum to fall from the tree to the ground.
We can use the equation for the distance fallen due to gravity:
d = 1/2 * g * t^2
where:
d = distance fallen (19.6 m)
g = acceleration due to gravity (9.8 m/s^2)
t = time
Let's solve for t:
19.6 = 1/2 * 9.8 * t^2
19.6 = 4.9 * t^2
t^2 = 19.6 / 4.9
t^2 = 4
t = √4
t = 2 seconds
Step 2: Calculate the acceleration of Sam's flat car.
We can use the equation for the distance traveled with constant acceleration:
d = 1/2 * a * t^2
where:
d = distance traveled (29.4 m)
a = acceleration (unknown)
t = time (2 seconds)
Let's solve for a:
29.4 = 1/2 * a * (2)^2
29.4 = 2 * a
a = 29.4 / 2
a = 14.7 m/s^2
Therefore, the acceleration of Sam's flat car to get him there quickly would be 14.7 m/s^2.
Step 3: Calculate the time it would require for Sam to catch the plum.
We can use the equation for the time it takes to travel a certain distance with constant acceleration:
d = 1/2 * a * t^2
where:
d = distance traveled (19.6 m)
a = acceleration (14.7 m/s^2)
t = time (unknown)
Let's solve for t:
19.6 = 1/2 * 14.7 * t^2
19.6 = 7.35 * t^2
t^2 = 19.6 / 7.35
t^2 = 2.6667
t = √2.6667
t = 1.63 seconds
Therefore, it would require approximately 1.63 seconds for Sam to catch the plum just before it hits the ground.
To calculate the acceleration of the flat car, we need to find the initial velocity of the plum when it starts falling. We can use the formula for free fall:
s = ut + 0.5 * a * t^2
where:
s = height of the plum (19.6 m)
u = initial velocity of the plum (0 m/s, as it is at rest)
a = acceleration of the plum (unknown)
t = time it takes for the plum to fall (unknown)
Since the plum is dropped vertically from rest, the initial velocity (u) is zero. Therefore, the equation simplifies to:
s = 0.5 * a * t^2
Plugging in the given height, we have:
19.6 = 0.5 * a * t^2
To find the acceleration (a), we need to isolate it in the equation. Rearranging the equation, we have:
a = (2s) / t^2
Now, to find the time it takes for the plum to fall, we need to use another formula of motion:
s = ut + 0.5 * a * t^2
But this time, the distance traveled is the horizontal distance (29.4 m) that Sam needs to cover to get directly under the falling plum, and the initial velocity (u) is zero. The equation then becomes:
29.4 = 0.5 * a * t^2
Rearranging the equation, we have:
a = (2s) / t^2
Now we have two equations with two unknowns (acceleration and time). Setting the two expressions for acceleration equal to each other, we find:
(2s) / t^2 = (2s) / t^2
This implies that the acceleration is the same in both cases. Therefore, we can simplify our calculations by equating the two equations:
19.6 = 29.4
This implies that the time it takes for the plum to fall is 1 second. Substituting this value back into one of the equations, we can find the acceleration:
19.6 = 0.5 * a * (1)^2
19.6 = 0.5 * a
a = 19.6 / 0.5
a = 39.2 m/s^2
So, the acceleration of the flat car needs to be 39.2 m/s^2 to quickly get Sam directly under the falling plum.
To calculate the time it requires for Sam to catch the plum, we have already determined that the time it takes for the plum to fall is 1 second. Therefore, it would require 1 second for Sam to catch the plum just before it hits the ground.