A ball of 250g hits the floor at a velocity of 2,50 m/s at an angle of 70* relative to the vertical. The vertical force in function with time between the floor and the ball is:
from 0 to 50 N : from 0 to 1 sec.
from 50 N to 100 N : from 1 to 2 sec.
constant 100 N : from 2 to 3 sec.
100 to 50 N : from 3 to 4 sec.
and 50 N to 0 N : from 4 to 5 sec.
What is the velocity of the ball after the collision if the ball bounces? Give the answer in terms of unit vectors and polar notation.
I know that I have to use 1/2 mv^2 for the velocity and mgh for gravitational energy, but I'm lost from here, especially when they ask to put it in unit vectors.
Thank you
To find the velocity of the ball after the collision, we need to consider the forces acting on the ball and use the principles of Newton's laws of motion.
First, let's break down the initial velocity of the ball into its vertical and horizontal components. Given that the velocity makes an angle of 70 degrees relative to the vertical, we can use trigonometry to find the vertical and horizontal velocities.
Vertical component: V_initial * sin(θ) = 2.50 m/s * sin(70°) = 2.3467 m/s
Horizontal component: V_initial * cos(θ) = 2.50 m/s * cos(70°) = 0.7660 m/s
Now, let's analyze the forces acting on the ball during the collision:
1. From 0 to 1 second: The vertical force is varying between 0 N and 50 N. As the force is not constant, we cannot directly calculate the change in velocity during this time. We would need additional information about the relationship between force and time.
2. From 1 to 2 seconds: The vertical force is 50 N, and the ball experiences a constant force of 50 N over this period. We can use Newton's second law (F = m * a) to find the acceleration (a = F / m) experienced by the ball during this time.
Acceleration (a) = 50 N / 0.250 kg = 200 m/s^2
Using the kinematic equation (v_final = v_initial + a * t), we can calculate the change in velocity during this time interval:
Change in velocity = 2.3467 m/s + 200 m/s^2 * (2 s - 1 s) = 402.3467 m/s
3. From 2 to 3 seconds: The vertical force is a constant 100 N. Similar to the previous step, we can calculate the acceleration during this time.
Acceleration = 100 N / 0.250 kg = 400 m/s^2
Change in velocity = 402.3467 m/s + 400 m/s^2 * (3 s - 2 s) = 802.3467 m/s
4. From 3 to 4 seconds: The vertical force is varying from 100 N to 50 N. Again, without knowing the relationship between force and time, we cannot calculate the exact change in velocity during this interval.
5. From 4 to 5 seconds: The vertical force is varying from 50 N to 0 N. Similarly, without the relationship between force and time, we cannot determine the change in velocity.
We have calculated the change in velocity during the intervals where the force is constant, but since we don't have enough information about the force during the non-constant intervals, we cannot determine the exact velocity of the ball after the collision.