A .25 kg skeet is fired at an angle of 30 degrees to the horizon with a speed of 30 m/s, When it reaches the maximum height, a 15 g pellet hits it from below. Before the collision, the pellet was traveling vertically upward at a speed of 200 m/s Embedded in the skeet, the two move together after the collision.
a) How much higher did the skeet go up?
b) How much distance does the skeet travel because of the collision?
For A-
m1v1 = m2v2
(.015kg + .25kg)V = .015kg + 200m/s
v = 11.3m/s
(1/2)mv^2 = mgh so h=(1/2)(v^2)/g
h = 6.5m
To solve this problem, we can use the principles of projectile motion and conservation of momentum.
a) To find how much higher the skeet goes up, we can use the fact that at the maximum height, the vertical component of the velocity is equal to zero.
The vertical component of the initial velocity can be found using the initial speed and the angle:
Vertical component of initial velocity = initial speed * sin(angle)
Vertical component of initial velocity = 30 m/s * sin(30°)
Vertical component of initial velocity = 15 m/s
To find the time it takes for the skeet to reach the maximum height, we can use the equation:
Final vertical velocity = initial vertical velocity - (acceleration due to gravity * time)
0 = 15 m/s - (9.8 m/s^2 * time)
9.8 m/s^2 * time = 15 m/s
time = 15 m/s / 9.8 m/s^2
time ≈ 1.53 s
To find how high the skeet goes up, we can use the equation for vertical displacement:
Vertical displacement = (initial vertical velocity * time) - (1/2 * acceleration due to gravity * time^2)
Vertical displacement = (15 m/s * 1.53 s) - (1/2 * 9.8 m/s^2 * (1.53 s)^2)
Vertical displacement ≈ 11.46 m
Therefore, the skeet goes up approximately 11.46 meters.
b) To find how much distance the skeet travels because of the collision, we need to calculate the horizontal component of the initial velocity.
The horizontal component of the initial velocity can be found using the initial speed and the angle:
Horizontal component of initial velocity = initial speed * cos(angle)
Horizontal component of initial velocity = 30 m/s * cos(30°)
Horizontal component of initial velocity = 25.98 m/s
The time the skeet takes to reach the maximum height can also be used as the time it takes to come back down.
To find the distance traveled, we can use the equation for horizontal displacement:
Horizontal displacement = horizontal component of initial velocity * time
Horizontal displacement = 25.98 m/s * 1.53 s
Horizontal displacement ≈ 39.81 m
Therefore, the skeet travels approximately 39.81 meters because of the collision.
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of mechanical energy.
a) To find out how much higher the skeet goes up, we need to calculate the maximum height of the skeet before the collision.
Let's first find the initial vertical velocity component of the skeet before the collision. We can use trigonometry to calculate this:
Vertical velocity component (v_skeet_y) = Velocity (v_skeet) * sin(angle)
v_skeet_y = 30 m/s * sin(30°)
v_skeet_y = 15 m/s
Next, we can calculate the maximum height (h_max) reached by the skeet using the conservation of mechanical energy:
Initial kinetic energy = Final potential energy at maximum height
(1/2) * mass_skeet * (v_skeet_y)^2 = mass_skeet * g * h_max
Substituting the given values:
(1/2) * 0.25 kg * (15 m/s)^2 = 0.25 kg * 9.8 m/s^2 * h_max
Simplifying the equation:
(1/2) * 225 = 2.45 * h_max
225 = 4.9 * h_max
h_max = 225 / 4.9
h_max ≈ 45.92 meters
Therefore, the skeet goes up approximately 45.92 meters.
b) To determine how much distance the skeet travels because of the collision, we need to find the horizontal velocity component of the skeet before the collision. We can again use trigonometry to calculate this:
Horizontal velocity component (v_skeet_x) = Velocity (v_skeet) * cos(angle)
v_skeet_x = 30 m/s * cos(30°)
v_skeet_x ≈ 25.98 m/s
Now, we can find the time it takes for the pellet to reach the skeet using the vertical velocity component of the pellet:
Vertical velocity component (v_pellet_y) = 200 m/s
v_pellet_y = -v_skeet_y (since the pellet is moving in the opposite direction to the skeet)
Using this equation, we can find the time the pellet takes to reach the skeet:
v_skeet_y = -v_pellet_y
15 m/s = 200 m/s * t
Simplifying the equation:
t = 15 m/s / 200 m/s
t = 0.075 seconds
Finally, we can calculate the horizontal distance (d_collision) traveled by the skeet due to the collision using the horizontal velocity component and the time:
d_collision = v_skeet_x * t
d_collision = 25.98 m/s * 0.075 s
d_collision ≈ 1.9495 meters
Therefore, the skeet travels approximately 1.9495 meters due to the collision.