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Unformatted text preview: he left, the angle between them must be 180 € degrees. So we can write the work due to the nonconservative force as W NC = f k cos(180 0 )Δx = − f k Δx Since the car stops, that tells us that the final velocity must be zero. So we can now write ΔKE = W NC 12 0 − mv 0 = − f k Δx 2 12 mv 0 = f k Δx 2
Solving for the displacement gives € 2 mv 0 Δx = = 76.56 m 2 fk € b) Since we know the distance to the truck, which is less than that of the stopping distance, the car will have a velocity when it hits the truck. Again due to the presence of the frictional force (which is the same as before) we can write ΔKE = W NC 1 21 2 mv − mv 0 = − f k Δx 2 2
Solving for the final velocity gives € 2 f k Δx v= v − = 27.3m / s m
2 0 Notes Example 2: Here we have to be careful, at the top of the jump the grasshopper has only the xcomponent of velocity. Since there is no acceleration in the xdirection we can write this velocity as
v x = v 0 x = v 0 cos( 45 0 ) = 2 v0 2 Since we are ignoring air resistance, this is a conservative system. So we can write € ΔE = ΔKE + ΔPE = 0
To find the change in potential energy, we define the ground as the y=0 point. Hence initial the grasshopper has zero potential energy. € We can now write ΔKE + ΔPE = 0 1 12 2 mv x − mv 0 + mgy = 0 2 2 2 12 1 2 m v 0 − mv 0 + mgy = 0 2 2 2 1 2 1 mv 0 − 1 + mgy = 0 2 2 1 2 − mv 0 + mgy = 0 4 € Solving for the initial velocity gives v 0 = 4 gy = 6.26 m / s € Extra Example 1: A box of mass m initially slides along a frictionless. It then runs into a spring, until the it momentarily stops. As the spring compresses the floor under the spring has a coefficient of kinetic friction. By what distance d is the spring compressed?
Friction Frictionless m = 2.0 kg k = 10000 N / m v 0 = 4.0 m / s f k = 15 N For the first part of the trip there is no friction force, thus it is a conservative system. After the box reaches the spring, the system is no longer a conservative s...
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 Fall '08
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