Problem 57 - College Physics

Question

In laboratory situations, a projectile's range can be used to determine its speed. To see how this is done, suppose a ball rolls off a horizontal table and lands \(1.5 \mathrm{~m}\) out from the edge of the table. If the tabletop is \(90 \mathrm{~cm}\) above the floor, determine (a) the time the ball is in the air, and (b) the ball's speed as it left the table top.

Step-by-Step Solution

Answer

(a) 0.428 seconds, (b) 3.505 m/s.

Step 1: Convert Units

First, convert the height of the table from centimeters to meters to be consistent with the units. Since there are 100 cm in a meter, the height of the table is \( 90 \text{ cm} = 0.9 \text{ m} \).

Step 2: Determine Time in Air

Use the formula for the time it takes an object to fall a certain height under gravity. The formula is \( t = \sqrt{\frac{2h}{g}} \), where \( h = 0.9 \text{ m} \) is the height and \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity. Substituting the values gives:\[t = \sqrt{\frac{2 \cdot 0.9}{9.8}} \approx \sqrt{\frac{1.8}{9.8}} \approx \sqrt{0.18367} \approx 0.428 \text{ s}\]

Step 3: Calculate Initial Speed

Now, use the horizontal range formula, \( R = v_i \cdot t \), where \( R = 1.5 \text{ m} \) is the distance from the table edge and \( t \) is the time in the air found previously. Solve for the initial speed as follows:\[v_i = \frac{R}{t} = \frac{1.5}{0.428} \approx 3.505 \text{ m/s}\]

Step 4: Interpret Results

The ball takes approximately \(0.428\) seconds to fall to the ground, and it must have left the table with a horizontal speed of approximately \(3.505\) m/s.

Key Concepts

Range of a Projectile

The range of a projectile refers to the horizontal distance it covers during its flight. In simpler terms, it's how far the object travels from the launch point before hitting the ground. In our example, the ball rolls off the table and lands 1.5 meters away. This 1.5 meters is what we identify as the 'range of the projectile'.
The range is importantly influenced by two significant factors:

• Initial speed
• Time of flight

These two topics are interconnected since the initial speed determines how far the projectile goes in the horizontal direction, while the time of flight dictates how long the projectile is in the air.
Understanding range is crucial for calculating other parameters like initial speed or time of flight when only the range is known.

Time of Flight

The time of flight of a projectile, like our ball, refers to the entire duration from the moment it leaves the launch point until it lands back on the ground. In the exercise, this is the amount of time the ball is airborne.
To determine the time of flight in such experiments, we can use the formula:\[t = \sqrt{\frac{2h}{g}} \]where:- \( t \) is the time of the flight,- \( h \) is the height from which it falls (converted to meters, in this case, 0.9 m), and- \( g \) is the gravity constant, approximately 9.8 m/s².
By plugging the values into the formula, the calculation progresses step-by-step, yielding around 0.428 seconds. Thus, it helps us link how long the projectile was in motion with other key factors.

Initial Speed Calculation

Determining the initial speed of a projectile involves understanding how fast it was moving when it first became airborne. This concept is pivotal because it connects directly to the projectile's range and time of flight.
In the given situation, with a known range (1.5 m) and time of flight (approximately 0.428 s), we utilize the basic kinematic equation:\[v_i = \frac{R}{t} \]where:- \( v_i \) is the initial speed we're solving for,- \( R \) is the range (1.5 m), and- \( t \) is the time of flight (0.428 seconds).
Substituting the values gives us an initial speed of roughly 3.505 m/s. Calculating this allows us to conclude how the factors of speed and time contribute to the distance traveled, demystifying the behavior of the projectile.