One strategy in a snowball fight is to throw a first...

One strategy in a snowball fight is to throw a first snowball at a high angle over level ground. While your opponent is watching the first one, you throw a second one at a low angle and timed to arrive at your opponent before or at the same time as the first one. Assume both snowballs are thrown with a speed of $25.0 \mathrm{~m} / \mathrm{s}$. The first one is thrown at an angle of $70.0^{\circ}$ with respect to the horizontal. (a) At what angle should the second (lowangle) snowball be thrown if it is to land at the same point as the first? (b) How many seconds later should the second snowball be thrown if it is to land at the same time as the first?

Key Concepts

Projectile Motion

Projectile motion involves the motion of an object that is launched into the air and is subject only to the acceleration of gravity. It requires analyzing both horizontal and vertical components of the motion separately using kinematics equations, and it provides the basis for determining parameters like range, maximum height, and time of flight.

Initial Velocity Decomposition

Decomposing the initial velocity into horizontal and vertical components is critical in projectile motion problems. The horizontal component (determined using the cosine of the launch angle) affects the range of the projectile, while the vertical component (determined using the sine of the launch angle) influences the time of flight and maximum height achieved.

Time of Flight

The time of flight is the total duration a projectile remains in the air before landing. It is dictated by the vertical component of the initial velocity and is calculated by considering the upward and downward journeys under gravity. This concept is essential when coordinating the timing of multiple projectiles to ensure simultaneous arrival.

Range Equation

The range of a projectile is the horizontal distance it covers before landing, which depends on both the initial speed and the launch angle. The range equation, derived from the kinematics of projectile motion, shows that for a given speed, complementary launch angles result in the same horizontal range, an important concept when planning trajectories that meet specific landing requirements.

Relative Launch Timing

When multiple projectiles are involved, coordinating the launch timing becomes crucial. Adjusting the launch delay between different projectiles ensures that they arrive at the target simultaneously, even when they are launched at different angles. This concept integrates the understanding of time of flight and ranges to achieve synchronized impacts.

Transcript

0:00 Hello.

00:02 This question speaks about a snowball fight and a specific strategy in the snowball fights is that you throw a first shot at a sharper angle.

00:16 So your opponent get distracted by this.

00:19 So let's assume that your opponent is here and you are staying at the origin.

00:26 So you throw the first one with angle and in this case it's 70.

00:32 So the angle here that it makes with the horizontal is 70.

00:38 And meanwhile, through another one at a lower angle so that it hits him directly.

00:48 And of course, the one here with lower angle will arrive faster than this one.

00:55 In this specific problem, we have that the initial velocity in both cases is 25 meters per.

01:04 Second and in the first part we're asked to get the angle at which you need to throw the second ball in order to rain to reach the same location and i can't answer this while drawing here and the answer should be 20 degree because that's the complementary of 70 both will add up to 90 degrees angle the actual reason for this is not of interest of for this problem, but if we see the equation of range, you will see that it's v .i squared sine 2 theta divided by g.

01:48 The interesting factor here is the 2 the sign of 2 theta because you can note here that sine 2 theta will be the same value for any two complementary angles.

02:04 So in case of 70, sign 140 is equal to sign 2 theta in case of 20, which is 40.

02:17 So this is from trig mathematics, and that's why complementary angles will land at the same position...

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