91Ó°ÊÓ

When an object is thrown horizontally, as with the firecracker in this scenario, its path is influenced by any forces acting parallel to the ground. In this exercise, a wind provides a constant horizontal acceleration, denoted as \(a\). This horizontal acceleration affects the firecracker's horizontal velocity.

This means that although the firecracker was initially thrown with a speed \(v\), it will experience an increase or decrease in that speed due to the wind's force. The formula to calculate the horizontal motion of the firecracker considered here is:

• \(x = vt + \frac{1}{2}at^2 \)

However, in this case, the firecracker ends up directly below where it was launched. This means the net horizontal displacement \(x\) is zero, demonstrating how effectively the wind counters the initial throw. Thus, understanding the force of horizontal acceleration is crucial in analyzing objects in projectile motion.

The time of flight is the total time an object remains in the air before landing. To calculate this for the firecracker, both the horizontal and vertical motions are considered. The key to solving this exercise is to realize that the time taken for vertical fall due to gravity is equal to the time for the horizontal motion affected by the wind.

Given that the horizontal displacement \(x\) is zero and using the equation updated for the force due to wind and throw \(x = vt + \frac{1}{2}at^2 = 0\), you can factor out \(t\) from the equation, leading to solutions that further explain the conditions affecting motion duration.

• \(t = -\frac{2v}{a} \)

While negative time is not feasible, the concept is key to bringing up the vertical motion perspective interrupted by gravity, allowing for complete understanding of equal time during descent.

Vertical motion in projectile scenarios considers how gravity influences an object's trajectory as it is falling. Initially, when the firecracker is thrown, it does not have an upward or downward speed component; hence its vertical velocity is zero. Instead, gravity \(g\) accelerates it downwards at \(9.8m/s^2\), which is crucial to determine the height.

The height \(h\) from which the firecracker falls can be found using the equation for designed free fall:

• \( h = \frac{1}{2}gt^2 \)

This equation reflects how long gravity influences the fall time. Since the time calculated has been equal for both horizontal and vertical directions, relating these concepts together can confirm the calculated height \(h\): \[h = \frac{2v^2}{ag}\]. This links back to how force and gravitational pull combine to dictate the firecracker's fall back to its launch origin point.