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!
V3 = (V1(M1-M2)+2M2*V2)/(M1+M2) = -0.169 m/s.
To solve this problem, start by considering the initial and final velocities of both gliders.
Given data:
- Mass of the first glider (m1) = 283 g = 0.283 kg
- Initial velocity of the first glider (v1) = 0.69 m/s
- Mass of the second glider (m2) = 467 g = 0.467 kg
- Final velocity of the second glider (v2) = ?
- Coefficient of kinetic friction (μk) = 0.02
- Length of the air track (L) = 2.4 m
Using the principle of conservation of momentum, m1v1 = m2v2, we can solve for v2:
0.283 kg × 0.69 m/s = 0.467 kg × v2
0.19527 kg∙m/s = 0.467 kg × v2
v2 = 0.19527 kg∙m/s ÷ 0.467 kg
v2 ≈ 0.418 m/s
After the collision, the second glider will move with a velocity of approximately 0.418 m/s.
Next, we need to consider the effect of friction on the second glider as it moves towards the end of the track.
The work done by friction is given by the equation W = μk × m2 × g × d, where:
- W is the work done by friction
- μk is the coefficient of kinetic friction
- m2 is the mass of the second glider
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- d is the distance traveled by the glider
The work done by friction will oppose the kinetic energy of the glider, so the work done by friction can be equated to the change in kinetic energy (KE) of the glider: W = ΔKE.
Using the equation for kinetic energy, KE = 0.5 × m × v², where m is the mass and v is the velocity, we can solve for ΔKE: ΔKE = KE_final - KE_initial.
Since the final velocity of the second glider (v2) is known, the kinetic energy of the second glider (KE_final) can be calculated as KE_final = 0.5 × m2 × v2².
The kinetic energy of the first glider (KE_initial) can be calculated as KE_initial = 0.5 × m1 × v1².
Therefore, ΔKE = (0.5 × m2 × v2²) - (0.5 × m1 × v1²).
Since the work done by friction (W) is equal to ΔKE, we can rewrite the equation as: W = (0.5 × m2 × v2²) - (0.5 × m1 × v1²).
Rearranging the equation to solve for the distance traveled by the second glider (d), we have: d = W ÷ (μk × m2 × g).
Now we can calculate the distance traveled by the second glider (d) using the known values of μk, m2, and g, which will determine if the glider reaches the end of the track.
Substituting the values,
d = (μk × m2 × g) ÷ W
d = (0.02 × 0.467 kg × 9.8 m/s²) ÷ (0.5 × 0.467 kg × 0.418 m/s²)
d ≈ 0.4 m
The second glider will travel approximately 0.4 m before coming to a stop due to friction. Since the length of the track is 2.4 m, 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 motion of the system before and after the collision, taking into account the friction force.
Let's start by calculating the initial momentum of the system. We have a 283 g (0.283 kg) air track glider moving at 0.69 m/s and a 467 g (0.467 kg) glider at rest. The mass and velocity of the glider at rest do not contribute to the initial momentum, so we only consider the moving glider:
Initial momentum of the system (before collision):
m1v1 = (0.283 kg)(0.69 m/s) = 0.19497 kg·m/s (rounding to 4 decimal places)
Now, let's analyze the collision using the principle of conservation of momentum. The collision is assumed to be elastic, which means the total momentum of the system is conserved.
Since the length of the gliders is neglected, we can consider only the collision between the moving glider and the stationary glider. After the collision, the moving glider will transfer some of its momentum to the stationary glider, causing it to move.
Let's denote the final velocities of the gliders as v1' and v2', respectively.
Applying the conservation of momentum:
m1v1 + m2v2 = m1v1' + m2v2'
Since m2v2 = 0 (the stationary glider), the equation simplifies to:
m1v1 = m1v1' + m2v2'
Now let's consider the friction force acting on the glider as it moves from the point of collision to the end of the track. The force of kinetic friction can be calculated using the equation:
friction force = coefficient of kinetic friction × normal force
The normal force acting on the glider can be determined by considering the weight of the glider, which can be calculated as:
weight = mass × gravitational acceleration
Since the gliders are on a horizontal air track, the normal force is equal to the weight of the glider (acting in the opposite direction).
Now, the friction force can be expressed as:
friction force = coefficient of kinetic friction × weight = coefficient of kinetic friction × mass × gravitational acceleration
Using this value, we can apply Newton's second law of motion to determine the acceleration caused by friction:
friction force = mass × acceleration
Now, we can determine the friction force acting on the moving glider.
Next, let's consider the motion of the glider after the collision. The net force acting on the glider can be calculated as the sum of the force of friction and the force causing acceleration due to the impulse.
The net force acting on the system is equal to the sum of the magnitudes of these two forces.
Using Newton's second law of motion:
net force = mass × acceleration
Since the acceleration is acting in the opposite direction of the glider's initial velocity, we can express the net force as:
net force = friction force − mass × acceleration
Now, we can calculate the net force acting on the glider.
Finally, we need to determine whether the glider will reach the end of the track. If the net force is greater than or equal to zero, the glider will eventually come to a stop before reaching the end of the track. If the net force is less than zero, the glider will continue to move and reach the end of the track.
Calculate the net force and evaluate whether it is greater than or equal to zero. If it is, the glider will not reach the end of the track. If it is less than zero, the glider will reach the end of the track.
M1*V1 + M2*V2 = M1*V3 + M2*V4.
0.283*V3 + 0.467*V4 = 0.195.
0.283*(-0.169) + 0.467*V4 = 0.195,
-0.048 + 0.467*V4 = 0.195,
V4 = (0.195+0.048)/0.467 = 0.520 m/s.
Fap*d = 0.5M2*V4^2.
0.283*9.8d = 0.5*0.467*0.52^2,
2.77d = 0.063,
d = 0.023 m.
0 = 0.52^2 - 2*5.75*d.
d = 0.0o24 m.