The velocity of a falling object in a vaccum is directly proportional to the amount of time the object has been falling. If after 5 seconds an object is falling at a speed of 90 miles per hour, how fast will it be falling after 12 seconds?

Answer

To solve the problem step by step, we will use the concept of direct proportionality between velocity and time.

### Step 1: Understand the relationship
The problem states that the velocity (v) of a falling object in a vacuum is directly proportional to the time (t) it has been falling. This can be expressed mathematically as:
\[ v = k \cdot t \]
where \( k \) is the constant of proportionality.

### Step 2: Find the constant of proportionality
We are given that after 5 seconds, the object is falling at a speed of 90 miles per hour. We can use this information to find \( k \):
\[ v = k \cdot t \]
Substituting the known values:
\[ 90 = k \cdot 5 \]
To find \( k \), divide both sides by 5:
\[ k = \frac{90}{5} = 18 \]

### Step 3: Write the equation for velocity
Now that we have \( k \), we can write the equation for velocity as:
\[ v = 18 \cdot t \]

### Step 4: Calculate the velocity after 12 seconds
We need to find the velocity after 12 seconds. Substitute \( t = 12 \) into the equation:
\[ v = 18 \cdot 12 \]
Calculating this gives:
\[ v = 216 \text{ miles per hour} \]

### Final Answer
The velocity of the object after 12 seconds will be **216 miles per hour**.

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