A projectile of mass 2.0 kg approaches a stationary target body at 4.9 m/s. The projectile is deflected through an angle of 59.8° and its speed after the collision is 2.8 m/s. What is the magnitude of the momentum of the target body after the collision?
To find the magnitude of the momentum of the target body after the collision, we will need to use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of a system remains constant before and after a collision, provided that no external forces act on the system. In this case, we can consider the projectile and the target body as the system.
Let's break down the problem into two components: one along the original path of the projectile (the x-axis) and one perpendicular to it (the y-axis).
Along the x-axis:
Before the collision, the projectile has a momentum of:
p₁ = mass × velocity = 2.0 kg × 4.9 m/s = 9.8 kg·m/s (taking the direction of motion as positive)
After the collision, the projectile changes its direction but we need to calculate its momentum along the x-axis afterward. To find it, we can use the velocity magnitude after the collision and the angle of deflection. We can use trigonometry to find the x-component of the velocity after the collision:
vx = v × cos(θ)
where v is the velocity magnitude after the collision and θ is the angle of deflection.
vx = 2.8 m/s × cos(59.8°) = 1.40 m/s
So the momentum of the projectile along the x-axis after the collision is:
p₂x = mass × velocity_x = 2.0 kg × 1.40 m/s = 2.80 kg·m/s (taking the direction of motion as positive)
Along the y-axis:
Since the target body is stationary, its momentum along the y-axis before and after the collision is zero (as there is no motion along the y-axis).
Now, we can calculate the momentum of the target body after the collision by considering the conservation of momentum.
In the x-direction:
Total initial momentum = Total final momentum
p₁ + 0 = p₂x + p_target
9.8 kg·m/s + 0 = 2.80 kg·m/s + p_target
Rearranging the equation, we can solve for the momentum of the target body:
p_target = 9.8 kg·m/s - 2.80 kg·m/s = 7.00 kg·m/s
Therefore, the magnitude of the momentum of the target body after the collision is 7.00 kg·m/s.