A 283 g air track glider moving at 0.69 m/s on a 2.4 m long air track collides elastically with a 467 g glider at rest in the middle of the horizontal track. The end of the track over which the struck glider moves is not frictionless, and the glider moves with a coefficient of kinetic friciton = 0.02 with respect to the track. Will the glider reach the end of the track? Neglect the length of the gliders.
So i know i will use m1v1=m2v2 but i don't know how to include the friction part, any help would be much appreciated!
To solve this problem, we can break it down into different steps:
Step 1: Calculate the initial momentum of the first glider before the collision.
The initial momentum of the first glider can be calculated using the formula: momentum = mass × velocity.
Momentum1 = (mass1 × velocity1) = (0.283 kg × 0.69 m/s) = 0.195 kgm/s
Step 2: Calculate the final momentum of the gliders after the collision.
Since the collision is elastic, we can equate the initial and final momentum values.
Momentum1 = Momentum2
Step 3: Calculate the velocity of the second glider after the collision.
Rearranging the equation from Step 2, we get:
velocity2 = (momentum1) / (mass2)
velocity2 = (0.195 kgm/s) / (0.467 kg) = 0.417 m/s
Step 4: Calculate the work done by friction on the second glider.
The work done by friction can be calculated using the formula: work = force × distance.
The force of friction can be calculated using the formula: force = coefficient of friction × normal force.
The normal force acting on the second glider is equal to its weight.
Normal force = mass2 × gravity
Normal force = (0.467 kg × 9.8 m/s^2) = 4.5696 N
The force of friction can be calculated as follows:
Force of friction = coefficient of friction × normal force
Force of friction = (0.02) × (4.5696 N) = 0.091392 N
Since friction acts over the distance traveled by the second glider, we can calculate the work done by friction using the formula:
Work = force of friction × distance
Work = (0.091392 N) × (2.4 m) = 0.21974112 J
Step 5: Calculate the change in kinetic energy of the second glider due to friction.
Change in kinetic energy = work done by friction
Change in kinetic energy = 0.21974112 J
Step 6: Calculate the final kinetic energy of the second glider.
The final kinetic energy can be calculated using the formula: KE = 0.5 × mass × velocity^2
Final kinetic energy = (0.5 × 0.467 kg) × (0.417 m/s)^2 = 0.05068305 J
Step 7: Determine if the glider reaches the end of the track.
For the glider to reach the end of the track, the final kinetic energy after the collision should be greater than the change in kinetic energy due to friction.
Final kinetic energy > Change in kinetic energy due to friction
0.05068305 J > 0.21974112 J
Since the final kinetic energy is less than the change in kinetic energy due to friction, the glider will not reach the end of the track.
To determine whether the glider will reach the end of the track, we need to analyze the forces acting on the gliders during the collision and their subsequent motion.
First, let's solve for the velocities of the gliders immediately after the collision using the principle of conservation of momentum:
m1v1 + m2v2 = m1v1' + m2v2'
where:
m1 = mass of the moving glider = 283 g = 0.283 kg
v1 = initial velocity of the moving glider = 0.69 m/s
m2 = mass of the glider at rest = 467 g = 0.467 kg
v2 = initial velocity of the glider at rest (which is 0 m/s)
v1' = final velocity of the moving glider (unknown)
v2' = final velocity of the glider at rest (unknown)
Using the given values, we have:
(0.283 kg)(0.69 m/s) + (0.467 kg)(0 m/s) = (0.283 kg)(v1') + (0.467 kg)(v2')
Simplifying the equation, we get:
0.195 kg·m/s = 0.283 kg·v1' + 0
So, we've determined the value of v1', the final velocity of the moving glider after the collision.
Now, we need to consider the subsequent motion of the gliders after the collision. The moving glider will now experience friction as it moves along the track. We can calculate the frictional force acting on the moving glider using the formula:
f_friction = μ * N
where:
μ = coefficient of kinetic friction = 0.02 (given)
N = normal force
Since the track is horizontal, the normal force N is equal to the weight of the glider, given by:
N = m1 * g
where:
m1 = mass of the moving glider = 0.283 kg
g = acceleration due to gravity = 9.8 m/s^2
Substituting the values, we have:
N = (0.283 kg)(9.8 m/s^2)
Now we can calculate the frictional force:
f_friction = (0.02)(0.283 kg)(9.8 m/s^2)
Having obtained the frictional force, we need to determine its effect on the motion of the glider. The frictional force acts in the direction opposite to the motion of the glider, so it will cause a deceleration.
Using Newton's second law, we can calculate the resulting deceleration:
f_friction = m1 * a
where:
m1 = mass of the moving glider = 0.283 kg
a = acceleration (deceleration in this case)
Solving for the deceleration, we have:
a = f_friction / m1
Now that we have the deceleration, we can determine the distance the glider will travel before coming to a halt. We can use the following equation:
v^2 = v0^2 + 2a * d
where:
v = final velocity of the glider (which is 0 m/s as it comes to a halt)
v0 = initial velocity of the glider = v1' (from the collision)
a = deceleration
d = distance traveled (unknown)
Rearranging the equation, we get:
d = (v^2 - v0^2) / (2a)
Substituting the values, we have:
d = (0 m/s)^2 - v1'^2 / (2a)
Since we have already determined the value of v1' from the collision calculation, we can substitute it into the equation to find the distance traveled.
Comparing this distance to the length of the track (2.4 m), if the distance traveled is less than 2.4 m, then the glider will come to a stop before reaching the end of the track. If the distance traveled is greater than or equal to 2.4 m, then the glider will reach the end of the track.
By following these steps, you should be able to determine whether the glider will reach the end of the track.