The acceleration due to gravity is -9.8 meters per second squared. An object is thrown upward from a 100-meter tower with a velocity of 49 meter per second.
a. Find the velocity function.
b. Find the position function
The correct answer and explanation is :
To solve this problem, we need to derive two functions for the object’s motion: the velocity function and the position function. We are given the following information:
- Initial velocity ( v_0 = 49 \, \text{m/s} )
- Initial position ( s_0 = 100 \, \text{m} )
- Acceleration due to gravity ( a = -9.8 \, \text{m/s}^2 )
a. Velocity Function
The velocity function ( v(t) ) describes how the velocity of the object changes over time. The acceleration ( a(t) ) is constant and is equal to ( -9.8 \, \text{m/s}^2 ). To find the velocity function, we integrate the acceleration function with respect to time.
The acceleration function is given by:
[
a(t) = -9.8 \, \text{m/s}^2
]
Since acceleration is the derivative of velocity, we integrate the acceleration function to find the velocity:
[
v(t) = \int a(t) \, dt = \int (-9.8) \, dt = -9.8t + C
]
To determine the constant ( C ), we use the initial condition that the velocity at ( t = 0 ) is ( v_0 = 49 \, \text{m/s} ):
[
v(0) = -9.8(0) + C = 49 \quad \Rightarrow \quad C = 49
]
Thus, the velocity function is:
[
v(t) = -9.8t + 49
]
b. Position Function
The position function ( s(t) ) describes the object’s height above the ground at any given time. The velocity function is the derivative of the position function, so we can integrate the velocity function to find the position function.
[
v(t) = \frac{ds}{dt} = -9.8t + 49
]
Now, integrate the velocity function to find the position function:
[
s(t) = \int v(t) \, dt = \int (-9.8t + 49) \, dt = -4.9t^2 + 49t + C_1
]
To determine the constant ( C_1 ), we use the initial condition that the position at ( t = 0 ) is ( s_0 = 100 \, \text{m} ):
[
s(0) = -4.9(0)^2 + 49(0) + C_1 = 100 \quad \Rightarrow \quad C_1 = 100
]
Thus, the position function is:
[
s(t) = -4.9t^2 + 49t + 100
]
Final Answer
- Velocity Function: ( v(t) = -9.8t + 49 )
- Position Function: ( s(t) = -4.9t^2 + 49t + 100 )
Explanation
- The velocity function ( v(t) = -9.8t + 49 ) shows that the object’s velocity decreases over time due to the negative acceleration caused by gravity. Initially, the object is moving upward at 49 m/s, but the velocity gradually decreases as it moves against gravity. After reaching the peak, the object will start descending.
- The position function ( s(t) = -4.9t^2 + 49t + 100 ) shows the object’s height above the ground at any given time. It is a quadratic equation, with the initial height of the object being 100 meters and a positive initial velocity. As time progresses, the object moves upward, slows down, and eventually starts falling back toward the ground. The negative coefficient of ( t^2 ) indicates that gravity is pulling the object downward after it reaches its peak height.