Question

One strategy in a snowball fight is to throw a snowball at a high angle over level ground. While your opponent is watching this first snowball, you throw a second snowball at a low angle and time it to arrive at the same time as the first. Assume both snowballs are thrown with the same initial speed 27.7 m/s. The first snowball is thrown at an angle of 53º above the horizontal. At what angle should you throw the second snowball to make it hit the same point as the first? Note the starting and ending heights are the same. The acceleration of gravity is 9.8 m/s^2. How many seconds after the first snowball should you throw the second so that they arrive on target at the same time? Answer in units of s."

Answer 1

To make both snowballs hit the same point at the same time, the angle for the second throw must be the complement of half the initial angle due to the symmetrical nature of projectile motion. The time for the second throw is determined by the time of flight, which is the same for both snowballs when thrown at the same speed but different angles for the same range.

Step-by-step explanation:

To determine the angle at which the second snowball should be thrown to ensure that both snowballs hit the same target at the same time, we need to consider the equations for projectile motion. Since both snowballs are thrown with the same initial speed and the only variable is the angle, the equation we are interested in involves the time of flight and the range of the projectile. The time of flight for a projectile thrown at an angle θ with an initial speed v is given by:

T = (2v · sin(θ)) / g

To hit the same target, both snowballs must have the same range. The range R of a projectile is given by:

R = (v^2 · sin(2θ)) / g

For the first snowball thrown at an angle of 53°:

R = (v^2 · sin(106°)) / g

Because the range is the same for both snowballs, this equation can be rearranged to solve for the angle of the second snowball, which needs to satisfy:

sin(2θ) = sin(106°)

This gives two possible angles, one acute and one obtuse. Since we are interested in the acute angle for the low throw, we subtract the first angle from 90 degrees to find the second angle:

θ_2 = (90° - (θ_1 / 2))

Now to find out how many seconds after the first snowball the second should be thrown, we calculate the time difference based on the property that both will have the same time of flight. We calculate the time of flight for the first snowball and then find the time at which the second snowball should be thrown to ensure they collide at the same point at the same time.