Problem 36 - Physics for Scientists and Engineers with Modern Physics

Question

Show that the time required for a projectile to reach its highest point is equal to the time for it to return to its original height if air resistance is neglible.

Step-by-Step Solution

Answer

The time to reach the highest point equals the time to return to the original height.

Step 1: Understanding the Problem

We need to analyze the motion of a projectile that is subject to gravity alone, without air resistance. The task is to demonstrate that the time it takes for the projectile to reach its peak is the same as returning to the launch point.

Step 2: Conceptual Framework

The motion of the projectile is described by uniformly accelerated motion. At the highest point, the projectile's vertical velocity becomes zero due to gravity decelerating its upward motion.

Step 3: Initial Equations

Express the initial conditions of the projectile. The initial velocity in the vertical direction is denoted as \( v_0 \). The acceleration due to gravity is \( g \), acting downward.

Step 4: Time to Reach Highest Point

Apply the equation \( v = v_0 - gt \) where \( v \) is the vertical velocity. Set \( v = 0 \) at the highest point to find the time \( t_1 \): \[ 0 = v_0 - g t_1 \]Solving for \( t_1 \), we get: \[ t_1 = \frac{v_0}{g} \]

Step 5: Time to Return to Original Height

At the original height, the projectile's vertical displacement is zero. The projectile will take twice the time to reach back the same level since the upward and downward journeys are symmetric. Therefore, the total time \( t_{total} \) is:\[ t_{total} = 2t_1 \]The time to return to the original height from the highest point is \( t_2 = t_1 \).

Step 6: Conclusion

Since \( t_1 \) is the time to reach the highest point, and the projectile takes \( t_1 \) again to return to the original height, we conclude that the ascent and descent times are equal when air resistance is negligible.

Key Concepts

Uniformly Accelerated Motion

In the realm of physics, uniformly accelerated motion refers to an object's movement when its velocity changes at a constant rate over time. This concept is crucial when studying projectile motion since gravity provides a constant acceleration downward.

In our exercise, consider a projectile launched into the air. The motion it undergoes is uniformly accelerated in the vertical direction due to gravity. This means:

• The velocity of the projectile will change steadily as it moves upwards and downwards.
• There is a consistent acceleration, in this case, due to gravity.

The equations of uniformly accelerated motion help us figure out things like time, velocity, and displacement at any point during the projectile's flight. Remember, the essence of uniformly accelerated motion is that it's predictable and constant, which means calculations become a bit simpler. The acceleration remains the same throughout, which in this situation is the gravitational pull acting only in one direction.

Vertical Velocity

Vertical velocity is the component of a projectile's velocity that acts in the vertical direction. It's crucial in understanding how high and how fast a projectile can go upwards before being pulled back down by gravity.

For projectile motion, the initial vertical velocity can be represented by the equation: \[ v = v_0 - gt \]Here:

• \( v \) is the velocity at any given time \( t \).
• \( v_0 \) is the initial vertical velocity at launch.
• \( g \) is the acceleration due to gravity.

As the projectile ascends, its vertical velocity decreases until it reaches the peak, where the vertical velocity becomes zero momentarily. This is due to gravity consistently decelerating the object. Once the peak is reached, the projectile starts to descend, and the vertical velocity increases again but in the direction opposite to the initial throw.

Knowing how vertical velocity changes throughout the flight helps predict the projectile’s trajectory and duration of flight.

Acceleration Due to Gravity

Acceleration due to gravity is a fundamental concept that influences the projectile's motion. It is a constant force that acts on the projectile to pull it back towards Earth. Understanding this concept is crucial as it dictates how a projectile behaves in the air.

Gravity's impact is significant because it:

• Applies a constant downward force on all objects regardless of their shape and mass (assuming negligible air resistance).
• Causes the projectile to decelerate as it moves upwards and accelerates as it comes back down.

For Earth, the acceleration due to gravity is approximately \( 9.8 \ m/s^2 \). This consistent force allows us to calculate various aspects of projectile motion precisely.

It's essential to remember that the symmetry of upward and downward journeys in projectile motion, as noted in the exercise, heavily relies on gravity being the only force acting on the projectile aside from its initial motion. This symmetry is what makes the ascent time equivalent to descent time when air resistance is not considered.