91Ó°ÊÓ

In physics, when an object moves at a constant velocity, it means that both its speed and direction remain unchanged over time. For our exercise, the cart moves horizontally with a constant velocity of 30.0 m/s to the right.

This implies that there is no acceleration acting in the horizontal direction. The horizontal motion of the cart can thus be described by the basic formula:

• \( d = v \times t \)

where \(d\) represents the distance traveled, \(v\) is the constant velocity, and \(t\) is the time.

Since there is no change in velocity, this simplifies the calculations we need to perform for the horizontal motion, enabling us to focus mainly on the vertical motion of the rocket.

A vertical launch in projectile motions refers to the scenario where an object is projected straight up into the air. In the exercise, our missile launcher propels the rocket upwards at an initial velocity of 40.0 m/s.

The vertical motion is governed by the principles of kinematics, and notably affected by gravity, which constantly acts downward. Thus, while calculating the rocket's maximum height or time of flight, only vertical velocity components and gravitational acceleration are considered.

The initial vertical launch velocity helps determine important metrics like:

• The maximum height using \(s = ut + \frac{1}{2} a t^2\).
• The time required to reach the peak height, where the upward velocity becomes zero.

This highlights the decoupling of vertical and horizontal motions, central in solving projectile problems.

Time of flight is a crucial concept in projectile motion, representing the total duration the object spends in the air, from the moment of launch until it returns to the same vertical level.

In our specific exercise, we determine the time it takes for the rocket to first reach its peak height, then double it to include the descending journey. The formula used is:

• \( v = u + at \)

where \(v\) is the final velocity at the peak (0 m/s), leading to the calculation.Once the peak's time is determined to be approximately 4.08 seconds, the total flight time becomes 8.16 seconds by multiplying the ascent time by two.

This entire duration considers the effect of gravity, with no external forces acting after the initial launch. Understanding this helps in synchronizing the concurrent horizontal and vertical motions.

Acceleration due to gravity is a constant force acting on projectiles, pulling them towards the Earth. Denoted by \(-9.8 \text{ m/s}^2\), it influences only the vertical motion of projectiles.

After the rocket's launch, gravity ensures that the initial vertical velocity decreases steadily until the peak is reached, where the velocity becomes zero. This point marks the highest point of the flight.

• During rise: Velocity decreases by 9.8 m/s every second.
• During descent: Velocity increases by 9.8 m/s every second back down.

Using gravity, we calculate maximum height and time of flight. Importantly, gravity acts independently of horizontal motion, simplifying calculations by allowing us to address vertical and horizontal components separately.