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Horizontal motion refers to the movement of an object across a horizontal plane. In the context of projectile motion, it is characterized by constant velocity. This is because, in a vacuum or an idealized scenario, no horizontal forces (like friction or air resistance) act to slow down or speed up the object. This is why the horizontal velocity remains unchanged throughout the motion.
In the example of the tiger leaping from the rock, the tiger's horizontal speed at launch is 3.2 m/s. This speed remains constant as there is nothing to accelerate or decelerate the tiger horizontally once she is airborne.

• Initial horizontal velocity = constant
• Horizontal distance = horizontal velocity × time of flight

It’s critical to separate horizontal and vertical motions in projectile problems to understand the complete trajectory.

Vertical motion under gravity is different from horizontal motion due to the presence of gravitational force, which causes acceleration toward the Earth. This means that the vertical velocity of an object changes over time. For this reason, it's essential to use kinematic equations that incorporate acceleration due to gravity.
In the tiger leap exercise, the tiger starts with zero vertical velocity because she only has a horizontal initial speed. However, gravity acts on her immediately, causing her to accelerate downward. This creates a parabolic trajectory overall.

• Initial vertical velocity = 0 (if only horizontal motion at the start)
• Acceleration due to gravity = 9.8 m/s² downwards
• Vertical distance covered depends on time

The formula used here to find the time taken to hit the ground is derived from the basic kinematic equation for uniformly accelerated motion: \[ h = \frac{1}{2}gt^2 \] where \(h\) is height and \(g\) is the acceleration due to gravity.

Kinematic equations are essential to solve problems involving motion, especially when the motion involves constant acceleration like gravity. These equations relate the five key parameters of motion:

• Initial velocity \( v_i \)
• Final velocity \( v_f \)
• Acceleration \( a \)
• Time \( t \)
• Displacement \( s \)

In the example, the formula for vertical motion utilized only time and height because the initial vertical velocity was zero. For horizontal motion, the appropriate kinematic equation was used: \[ d = vt \]This equation works without modifications since the horizontal velocity is constant, essentially equating the horizontal displacement \(d\) to the product of velocity and time.

The time of flight in projectile motion is the total time an object spends in the air from launch to landing. It’s one of the crucial parameters for determining the horizontal range of the projectile.
To solve for the time of flight when an object is launched horizontally, we only need to focus on its vertical motion since horizontal motion has no influence over the time it takes to hit the ground. In the tiger’s leap, this time was determined by the height of the rock and the acceleration due to gravity:
\[ h = \frac{1}{2}gt^2 \] The tiger’s time of flight calculated in the solution was approximately 1.24 seconds, which is a typical scenario when objects only have gravitational force acting on their vertical movement. This time is then utilized to calculate how far it travels horizontally.

• Time calculated from vertical motion
• Influenced by gravity and not initial horizontal velocity

Understanding time of flight provides insight into how far or how long an object will travel before coming to a stop.