A catapult throws a projectile at a 40.0° angle with a velocity of 26.0 m/s. How high will the projectile reach, and how far away will the projectile land?
u =horizontal speed = 26 cos 40 = 19.9 m/s
Vi initial vertical = 26 sin 40 = 16.7 m/s
time to top:
v = 0 = Vi - 9.8 t
9.8 t = 16.7
t = 1.705 s to top (3.41 s total in air)
HOW HIGH?
h = 0 + Vi t -4.9 t^2
= 16.7(1.705) - 4.9 (1.705)^2
= 28.47 - 14.25
= 14.2 meters
HOW FAR?
u * time in air = 19.9*3.41 = 67.9 m
A super-hero is leaping from the roof of one building onto another. The first building is 120 meters high and the second building is 80 meters high. The horizontal velocity of the super-hero is 12.0 m/s. What is the maximum possible distance between buildings that the hero can leap across?
To solve this problem, we can use the equations of projectile motion. The vertical and horizontal motions of the projectile are independent of each other, so we can analyze them separately.
1. Finding the vertical motion:
Since the projectile is launched at an angle of 40.0°, we need to decompose the initial velocity into its vertical and horizontal components. The vertical component can be found using the formula:
Vertical component of velocity (Vy) = Initial velocity (V) × sin(θ)
Vy = 26.0 m/s × sin(40.0°) = 16.75 m/s
Now, to find the height reached by the projectile, we can use the formula for vertical displacement of a projectile:
Vertical displacement (Δy) = (Vy^2) / (2 × acceleration due to gravity)
Acceleration due to gravity is approximately 9.8 m/s^2 (assuming no air resistance).
Δy = (16.75 m/s)^2 / (2 × 9.8 m/s^2) ≈ 14.32 m
So, the projectile will reach a height of approximately 14.32 meters.
2. Finding the horizontal motion:
The horizontal component of the initial velocity remains constant throughout the motion. We can find the horizontal component using the formula:
Horizontal component of velocity (Vx) = Initial velocity (V) × cos(θ)
Vx = 26.0 m/s × cos(40.0°) = 19.85 m/s
To find the horizontal displacement (range), we need to calculate the time of flight (T). The time of flight can be determined using the formula:
Total time of flight (T) = (2 × Vy) / acceleration due to gravity
T = (2 × 16.75 m/s) / 9.8 m/s^2 ≈ 3.43 s
Now, we can calculate the horizontal displacement using the formula:
Horizontal displacement (range) = Vx × T
Range = 19.85 m/s × 3.43 s ≈ 68.11 m
Therefore, the projectile will land approximately 68.11 meters away from the catapult.