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.
and just a theory question:
When I'm asked to find work done by air resistance against a skier on a slope, will I have to do:
Wr = -Fr*d
or is it affected by an angle:
Wr = -Fr*d*cosTHETA
I was thinking that it isn't because air resistance isn't a specific force acting at one spot but everywhere on the slope at the same time, unless it's what's acting on the skier himself, then I would have to add the angle into my calculations
First question:
The sum of the force*time products will tell you the total impulse in the vertical direction, from which the final vertical velocity can be calculated (by dividing by mass). It will tell you nothing about the impulse in the horizontal direction, which is not necessarily zero. The question is poorly conceived.
Second question:
(Air resistance force) * d, because that force will be opposite to the direction of motion, regardless of the hill angle.
thank you very much!
Actually the direction of the air rersistance force can be somewhat different from the direction opposite to motion, depending upon the body's inclination to the direction of motion. Ski jumpers take advantage of this by leaning forward and tilting their ski tips up to stay aloft longer.
but the angle will still have no effect right?
To determine the velocity of the ball after the collision, we can use the law of conservation of energy and momentum.
First, let's analyze the forces acting on the ball during the collision. The vertical force can be broken down into two components: the force perpendicular to the floor (normal force) and the force parallel to the floor (friction force). However, since there is no information about the coefficient of restitution or the characteristics of the floor, we will assume an ideal elastic collision where there is no loss of energy.
Using the given information, we can determine the vertical force acting on the ball in different time intervals:
From 0 to 1 second (0 to 50 N):
The vertical force increases from 0 to 50 N.
From 1 to 2 seconds (50 to 100 N):
The vertical force further increases from 50 N to 100 N.
From 2 to 3 seconds (constant 100 N):
The vertical force remains constant at 100 N.
From 3 to 4 seconds (100 to 50 N):
The vertical force decreases from 100 N to 50 N.
From 4 to 5 seconds (50 to 0 N):
The vertical force decreases from 50 N to 0 N.
To find the velocity after the collision, we need to determine the acceleration and time intervals corresponding to each force change. With this information, we can integrate the acceleration function to obtain the velocity function.
Assuming uniform acceleration during each time interval, we can calculate the changes in velocity (∆v) for each interval:
From 0 to 1 second:
Acceleration = (50 N - 0 N) / (1 s - 0 s) = 50 m/s^2
∆v = acceleration * ∆t = 50 m/s^2 * 1 s = 50 m/s
From 1 to 2 seconds:
Acceleration = (100 N - 50 N) / (2 s - 1 s) = 50 m/s^2
∆v = acceleration * ∆t = 50 m/s^2 * 1 s = 50 m/s
From 2 to 3 seconds:
Acceleration = 0 m/s^2 (constant force)
∆v = acceleration * ∆t = 0 m/s^2 * 1 s = 0 m/s
From 3 to 4 seconds:
Acceleration = (50 N - 100 N) / (4 s - 3 s) = -50 m/s^2 (negative because it is in the opposite direction)
∆v = acceleration * ∆t = -50 m/s^2 * 1 s = -50 m/s
From 4 to 5 seconds:
Acceleration = (0 N - 50 N) / (5 s - 4 s) = -50 m/s^2 (negative because it is in the opposite direction)
∆v = acceleration * ∆t = -50 m/s^2 * 1 s = -50 m/s
Now, let's sum up the changes in velocity (∆v) for each interval:
∆v_total = ∆v_0-1 + ∆v_1-2 + ∆v_2-3 + ∆v_3-4 + ∆v_4-5
∆v_total = 50 m/s + 50 m/s + 0 m/s - 50 m/s - 50 m/s
∆v_total = 0 m/s
The total change in velocity (∆v_total) is 0 m/s, meaning the ball comes to rest after the collision. Therefore, the velocity of the ball after the collision in terms of unit vectors and polar notation is 0 m/s.
Regarding your theory question about work done by air resistance, it depends on how you define the direction of the displacement (d). If the displacement is defined along the direction of motion, you would use:
Wr = -Fr * d
where Fr is the force of air resistance and d is the displacement along the direction of motion. The negative sign indicates that the work done by air resistance is in the opposite direction of the displacement.
On the other hand, if you want to account for the angle (θ) between the force of air resistance and the displacement, you would use:
Wr = -Fr * d * cosθ
This equation incorporates the angle θ to adjust for the portion of the force acting in the direction of the displacement. If the force and displacement are perpendicular (θ = 90°), its cosine will be zero, resulting in no work being done by air resistance in that direction.
Overall, you need to consider the specific scenario and define the appropriate displacement direction and angle to accurately calculate the work done by air resistance.