A tomato is thrown upward from a bridge 23 m above the ground at 40 m/sec. find acceleration, velocity, and height of tomato at time t. How high does the tomato go, and when does it reach its highest point?
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To find the acceleration, velocity, and height of the tomato at time t, we can use the equations of motion.
1. Acceleration (a): The acceleration in this case is constant and is equal to the acceleration due to gravity, which is approximately 9.8 m/s² (assuming no air resistance).
2. Initial velocity (u): The tomato is thrown upward from the bridge with an initial velocity of 40 m/s. Since it is going upward, the initial velocity is positive.
3. Time (t): We need to determine the time at which we want to find the acceleration, velocity, and height of the tomato.
4. Using the equation of motion for displacement: s = ut + (1/2)at²
- We know the initial displacement (s) is 23 m (the height of the bridge above the ground).
- The initial velocity (u) is 40 m/s.
- The acceleration (a) is -9.8 m/s² (taking downward direction as negative).
- The time (t) is the variable we need to solve for.
Let's solve for time (t) first:
23 = (40)t + (1/2)(-9.8)t²
23 = 40t - 4.9t²
4.9t² - 40t + 23 = 0
Using the quadratic formula, t = (-b ± √(b² - 4ac)) / (2a), where
a = 4.9, b = -40, and c = 23.
Solving for t, we find two possible values. We will consider the positive value (as time can't be negative):
t ≈ 0.94 seconds
Now, let's calculate the values of acceleration, velocity, and height of the tomato at time t = 0.94 seconds.
Acceleration (a): The acceleration remains constant throughout the motion and is equal to 9.8 m/s² (downward direction).
Velocity (v): The velocity at time t is given by v = u + at, where
u = 40 m/s (initial velocity) and
a = -9.8 m/s² (acceleration).
v = 40 - 9.8 * 0.94
v ≈ 30.28 m/s (upward)
Height (h): The height at time t is given by h = s + ut + (1/2)at², where
s = 23 m (initial height).
h = 23 + 40 * 0.94 + (1/2) * -9.8 * (0.94)²
h ≈ 43.48 m
Therefore, at time t = 0.94 seconds, the tomato has an acceleration of approximately -9.8 m/s² (downward), a velocity of about 30.28 m/s (upward), and a height of around 43.48 m above the ground.
To find the highest point reached by the tomato, we need to determine the time it takes for the tomato to reach its highest point. At this point, the velocity will be zero.
Using the equation v = u + at and considering the upward direction as positive:
0 = 40 - 9.8t
9.8t = 40
t ≈ 4.08 seconds
Thus, the tomato will reach its highest point at approximately 4.08 seconds, and its highest height will be calculated by substituting this time into the equation of motion for height.