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 32 m/s. The first snowball is thrown at an angle of 52◦ 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/s2
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.
To find the angle at which the second snowball should be thrown, we can use the following steps:
Step 1: Find the horizontal and vertical components of the first snowball's velocity.
Given that the initial speed is 32 m/s and the angle of projection is 52 degrees, we can find the horizontal component (Vx1) and vertical component (Vy1) of the velocity using the following equations:
Vx1 = V1 * cos(θ1)
= 32 * cos(52°)
Vy1 = V1 * sin(θ1)
= 32 * sin(52°)
Step 2: Find the time taken by the first snowball to reach the target position.
Since the starting and ending heights are the same, the vertical displacement (Δy1) of the first snowball is zero. Using the equation of motion:
Δy1 = Vy1 * t1 + (1/2) * g * t1^2
0 = (32 * sin(52°)) * t1 + (1/2) * (9.8) * t1^2
This equation is a quadratic equation in terms of t1. We can solve it using the quadratic formula to find the time taken by the first snowball.
Step 3: Find the horizontal distance traveled by the first snowball.
Using the equation of motion for horizontal motion:
Δx1 = Vx1 * t1
= (32 * cos(52°)) * t1
Step 4: Find the time at which the second snowball should be thrown to arrive at the same time.
Since the second snowball is thrown later, the time taken by the second snowball will be t1 + Δt, where Δt is the time difference between the throws. Using this information, we can write:
t2 = t1 + Δt
Step 5: Find the angle at which the second snowball should be thrown.
We want the second snowball to hit the same point as the first, so the horizontal displacement of the second snowball (Δx2) should be equal to Δx1. Using the equation of motion for horizontal motion:
Δx2 = (Vx2 * t2)
= (V2 * cos(θ2)) * (t1 + Δt)
Since Δx2 = Δx1, we can equate the two expressions:
(32 * cos(52°)) * t1 = (V2 * cos(θ2)) * (t1 + Δt)
Solving this equation for θ2 will give us the angle at which the second snowball should be thrown.
Finally, to find how many seconds after the first snowball should you throw the second so that they arrive on target at the same time, you need to calculate Δt, the time difference between the throws. Depending on the specific details of the problem, this information might be given or you would need additional equations or data to determine this time difference.