One strategy in a snowball fight is to throw

Question

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.
The acceleration of gravity is 9.8 m/s2 .
Assume both snowballs are thrown with
the same initial speed 35.2 m/s. The first
snowball is thrown at an angle of 51� above
the horizontal. At what angle should you
throw the second snowball to make it hit the
same point as the first?
Answer in units of �.

Answer 1

We can solve this problem by breaking down the motion of the two snowballs into horizontal and vertical components.

Let's focus on the first snowball thrown at a 51° angle. The initial velocities for the first snowball in each direction are:

Vx1 = 35.2 * cos(51°)
Vy1 = 35.2 * sin(51°)

Now, we need to find the time it takes for the first snowball to reach its target. To do this, we will use the following equation for the vertical motion (since the second snowball will have the same time in reaching the target):

y = Vy1 * t - (1/2) * g * t^2

Since the snowball is thrown and lands at the same height, the vertical displacement (y) will be zero:

0 = Vy1 * t - (1/2) * g * t^2

Rearranging to solve for t:

t = (2 * Vy1) / g

Now, let's find the horizontal distance (x) traveled by the first snowball:

x = Vx1 * t

The second snowball is thrown at a lower angle θ. The initial velocities for the second snowball in each direction are:

Vx2 = 35.2 * cos(θ)
Vy2 = 35.2 * sin(θ)

The horizontal distance traveled by the second snowball is the same as the first snowball:

x = Vx2 * t

Since we know that t is the same for both snowballs, we can set the two horizontal distance equations equal to each other:

Vx1 * t = Vx2 * t

Canceling the t on both sides:

Vx1 = Vx2

We can now substitute the values of Vx1 and Vx2 in terms of the two angles:

35.2 * cos(51°) = 35.2 * cos(θ)

Since the initial speed is the same for both snowballs, we can divide both sides by 35.2 to simplify:

cos(51°) = cos(θ)

Now, to solve for θ:

θ = arccos(cos(51°))

θ ≈ 39°

Therefore, you should throw the second snowball at a 39° angle to make it hit the same point as the first snowball.