A block is initially sliding along a horizontal surface with 24.6 J of kinetic energy. Friction causes the block’s speed to be halved in a distance of 3.20 m.
a) What is the kinetic energy of the block after traveling the 3.20 m?
b) What is the magnitude of the kinetic frictional force that acts on the block?
c) How much further will the block travel before coming to a complete stop?
(a)KE1=m•v1²/2=24.6 J.
v2=v1/2
KE2= m•v2²/2 =
=m•v1²/2•4=KE1/4=
=24.6/4=6.15 J.
(b)W(fr)=KE2-KE1=
6.15-24.6 = -18.45 J.
W(fr)=F(fr) •s•cos180º=>
F(fr) = W(fr)/ s•cos180º=
=-18.45/3.2•(-1)=5.77 N.
(c) KE2=- W(fr1)= - F(fr) •s1•cos180º
s1 = KE2/ F(fr)•cos180º=
=6.15/(-5.77) •(-1)=1.07 m.
To solve this problem, we need to use the principle of work and energy.
Given:
Initial kinetic energy (KE₁) = 24.6 J
Distance traveled (d) = 3.20 m
a) To find the kinetic energy of the block after traveling the 3.20 m, we can use the work-energy theorem:
Work (W) = Change in kinetic energy (ΔKE)
The work done by friction is equal to the change in kinetic energy:
W = -ΔKE
Since the block's speed is halved, the final kinetic energy (KE₂) will be half of the initial kinetic energy:
KE₂ = (1/2) * KE₁
Substituting the given values:
KE₂ = (1/2) * 24.6 J
KE₂ = 12.3 J
Therefore, the kinetic energy of the block after traveling the 3.20 m is 12.3 J.
b) To find the magnitude of the kinetic frictional force that acts on the block, we can use the work-energy theorem again:
W = -ΔKE
The work done by friction is equal to the force of friction multiplied by the displacement:
W = F_friction * d
Since the work done by friction is negative, the magnitude of the force of friction will be:
|F_friction| = |W / d|
Substituting the given values:
|F_friction| = |24.6 J / 3.20 m|
|F_friction| = 7.69 N
Therefore, the magnitude of the kinetic frictional force that acts on the block is 7.69 N.
c) To find how much further the block will travel before coming to a complete stop, we need to find the distance it will travel while losing all its kinetic energy.
The work done by friction is equal to the force of friction multiplied by the displacement:
W = F_friction * d'
Since the block comes to a complete stop, the final kinetic energy is zero:
W = -ΔKE = -KE
Substituting the values:
W = -24.6 J
|F_friction| * d' = -24.6 J
|F_friction| = 7.69 N (from part b)
7.69 N * d' = -24.6 J
d' = (-24.6 J) / (7.69 N)
d' ≈ -3.20 m
The negative sign indicates that the block will travel 3.20 m in the opposite direction before coming to a complete stop.
Therefore, the block will travel an additional 3.20 m in the opposite direction before coming to a complete stop.
To answer these questions, we need to apply principles of energy and work. Let's break it down step by step:
a) To find the kinetic energy of the block after traveling 3.20 m, we need to know the work done by the frictional force. The work done by a force is given by the equation W = Fd, where W is work, F is force, and d is distance. Since the block's speed is halved, we know that the work done by friction is equal to the change in kinetic energy.
The initial kinetic energy of the block is given as 24.6 J. Since the speed is halved, the final kinetic energy will be half of the initial kinetic energy.
Therefore, the kinetic energy of the block after traveling the 3.20 m is 24.6 J / 2 = 12.3 J.
b) To find the magnitude of the kinetic frictional force, we can use the work-energy principle again. The work done by friction is equal to the force of friction multiplied by the distance.
The work done by friction is the change in kinetic energy, which we found to be 12.3 J. The distance traveled is given as 3.20 m.
Therefore, the magnitude of the kinetic frictional force can be calculated by rearranging the equation W = Fd as F = W / d.
Thus, the magnitude of the kinetic frictional force is 12.3 J / 3.20 m = 3.84 N.
c) To find how much further the block will travel before coming to a complete stop, we can use the work-energy principle once again. The work done by friction causes the change in kinetic energy, and this work is equal to the force of friction multiplied by the distance.
We know the work done by friction is equal to the initial kinetic energy of the block, which is 24.6 J. We also know the magnitude of the kinetic frictional force is 3.84 N.
Using the equation W = Fd, we can rearrange it to find the distance, d = W / F.
Thus, the distance the block will travel before coming to a complete stop is 24.6 J / 3.84 N = 6.41 m.
Therefore, after traveling the 3.20 m, the block will have a kinetic energy of 12.3 J, the magnitude of the kinetic frictional force is 3.84 N, and it will travel an additional 6.41 m before coming to a complete stop.