So we can write the work due to the non conservative

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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 non-conservative 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 x-component of velocity. Since there is no acceleration in the x-direction 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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