At the end of the spring term, a high school physics class celebrates by shooting a bundle of exam papers into the town landfill with a homemade catapult. They aim for a point that is 31.1 m away and at the same height from which the catapult releases the bundle. The initial horizontal velocity component is 5.87 m/s.
a) What is the initial velocity component in the vertical direction?
To find the initial velocity component in the vertical direction, we can use the concept of projectile motion. In this case, the horizontal and vertical motions are independent of each other.
Given that the horizontal velocity component, Vx, is 5.87 m/s, we need to find the initial vertical velocity component, Vy.
The initial velocity of the projectile (V) can be found using the Pythagorean theorem, which states that the square of the magnitude of the velocity (V) is equal to the square of its horizontal component (Vx) plus the square of its vertical component (Vy).
Mathematically, it can be represented as:
V^2 = Vx^2 + Vy^2
Rearranging the equation, we get:
Vy^2 = V^2 - Vx^2
Now, we have the value of Vx (5.87 m/s). However, we need to find the value of V. But since the problem doesn't provide the value of V directly, we need to find it using other known information.
In the problem statement, it is mentioned that the point the students aim for is 31.1 m away and at the same height from which the catapult releases the bundle. From this, we can infer that the horizontal travel distance (range) and the initial height (h) are known.
The range of a projectile (R) can be calculated using the formula:
R = (Vx * t)
where t is the time of flight.
Also, the time of flight (t) can be calculated using the formula:
t = (2 * Vy) / g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, we can substitute the given range value and solve for t:
31.1 m = (5.87 m/s) * t
t = 31.1 m / 5.87 m/s
t ≈ 5.30 s
Now, we can substitute the calculated value of t into the formula for Vy:
Vy = (g * t) / 2
Vy = (9.8 m/s^2) * (5.30 s) / 2
Vy ≈ 25.97 m/s
Therefore, the initial velocity component in the vertical direction (Vy) is approximately 25.97 m/s.