91Ó°ÊÓ

Calculating velocity is a key part of kinematics, allowing us to understand how the speed of an object changes over time. When dealing with a problem like the one presented, where a rock is launched upward, we use the kinematic equation:

• \[ v = u + at \]

Here,

• \( v \) is the final velocity,
• \( u \) is the initial velocity,
• \( a \) is the acceleration, and
• \( t \) is the time elapsed.

This equation helps us calculate the velocity at different points in time, assuming that the acceleration (such as gravity) is constant. Velocity can be both positive and negative. Positive means the object is moving upward, and negative means it is moving downward. This approach is essential in physics problems to predict how objects behave when forces like gravity act on them.
In our original exercise, we see this application by calculating both the initial velocity and the velocity after the rock has been in motion for 5 seconds.

Acceleration due to gravity is a fundamental concept in physics, especially when dealing with motion problems. On Earth, the acceleration due to gravity is a constant \( -9.8 \text{ m/s}^2 \). The negative sign indicates that gravity acts downward, pulling objects towards the Earth.
When an object is launched upward, like the rock in our exercise, gravity works against the initial velocity. This means every second, the velocity of the rock decreases by \( 9.8 \text{ m/s} \) until it comes to a momentary stop at its peak and then begins to fall back down.
Understanding this concept allows us to predict the behavior of objects under gravity accurately. It's why, in the solution, the velocity of the rock becomes negative at \( t = 5.0 \text{ s} \); the rock is moving downward, having been unable to overcome the force of gravity acting on it throughout its flight.

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. This results in a parabolic path under ideal conditions (like ignoring air resistance).
For the rock in our exercise, its motion can be split into two phases:

• Rising up until its velocity becomes zero, and
• Then falling back down to the ground.

The time taken to reach the highest point is crucial in understanding the total flight time of the projectile. By using the kinematic equations, we find the velocities at different times, allowing us to outline its entire trajectory. This gives deeper insights into the effects of initial forces and gravity's pull.
Such problems also illustrate that projectile motion is symmetrical. The time to reach the peak is equal to the time taken to fall from that peak back to the original launch height, assuming no other forces act. Understanding these principles can help predict and calculate future motion behaviours of projectiles accurately.