A 0.25 kg skeet (clay target) is fired at an angle of 30 degrees to the horizon with a speed of 25 m/s. When it reaches the maximum height, it is hit from below by a 15 g pellet travelling upward at a speed of 200 m/s. The pellet is embedded in the skeet. (a) How much higher did the skeet go up? (b) How much extra distance does the skeet travel because of the collision?
To calculate the answers to these questions, we need to apply principles of projectile motion and conservation of momentum.
(a) To determine how much higher the skeet goes up after the collision, we first need to find the maximum height it reaches before the collision. We can use the kinematic equation for projectile motion relating the vertical displacement (Δy), initial velocity (viy), and acceleration due to gravity (g).
Δy = (viy^2) / (2g)
Given that the skeet is fired at an angle of 30 degrees to the horizon, we can split the initial velocity into its horizontal (vix) and vertical (viy) components. The horizontal velocity remains constant throughout the motion, while the vertical velocity changes due to the acceleration of gravity.
vix = vicosθ
viy = visinθ
Where vi is the initial speed of the skeet (25 m/s) and θ is the launch angle (30 degrees).
Plug in the values:
vix = 25 m/s * cos(30 degrees)
viy = 25 m/s * sin(30 degrees)
Next, calculate the time taken to reach the maximum height. At the maximum height, the vertical velocity becomes zero (vy = 0). We can use the kinematic equation:
vy = viy - gt
Rearranging the equation to solve for time (t):
t = viy / g
Then, substitute the value:
t = (25 m/s * sin(30 degrees)) / (9.8 m/s^2)
Finally, use this time to find the vertical displacement of the skeet:
Δy = (viy^2) / (2g)
Δy = (25 m/s * sin(30 degrees))^2 / (2 * 9.8 m/s^2)
This gives you the additional height the skeet reaches after the collision.
(b) To calculate the extra distance traveled by the skeet due to the collision, we need to consider conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
The momentum of the skeet before the collision is given by:
pskeet = (mass of skeet) * (vertical velocity of the skeet before collision)
The momentum of the pellet before the collision is given by:
ppellet = (mass of pellet) * (vertical velocity of the pellet before collision)
The total momentum before the collision is:
ptotal_before = pskeet + ppellet
The total momentum after the collision is:
ptotal_after = (mass of skeet + mass of pellet) * (vertical velocity of the combined masses after collision)
Since the pellet is embedded in the skeet after the collision, the vertical velocity of the combined masses after collision can be calculated using conservation of momentum:
(mass of skeet + mass of pellet) * (vertical velocity of the combined masses after collision) = ptotal_before
Rearranging the equation, we can solve for the vertical velocity of the combined masses after the collision.
Finally, we can calculate the distance traveled by the skeet after the collision using the horizontal velocity:
distance = vix * time
Plug in the values obtained earlier to find the extra distance traveled by the skeet due to the collision.
Note: Make sure to convert the masses from grams to kilograms before performing calculations.