Collisions and Explosions

  • In the case of a collision or explosion, if you add up the individual momentum vectors of all the objects before the event, you'll find that they are equal to the sum of momentum vectors of the objects after the event

  • Written mathematically, the law of conservation of linear momentum states

  • C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image130.png

Solving Momentum Problems

  1. Identify all the objects in the system

  2. Determine the momenta of the objects before the event. Use variables for any unknowns

  3. Determine the momenta of the objects after the event. Use variables for any unknowns

  4. Add up all the momenta from before the event and set equal to the momenta after the event

  5. Solve for any unknowns

Types of Collisions

Elastic collisi on Objects that collide move separately after collision. Inelastic collision ee Objects that collide move together after collision. Total momentum and total energy of the system are conserved. Kinetic energy is conserved. Kinetic energy is NOT conserved.

  • Elastic collision

    • Kinetic energy is conserved

  • Inelastic collision

    • Kinetic energy is not conserved

Example 1: Traffic Collision

  • A 2000-kg car traveling 20 m/s collides with a 1000-kg car at rest. If the 2000-kg car has a velocity of 6.67 m/s after the collision, find the velocity of the 1000-kg car after the collision

    ф5есЬ ты R (97) ?то.го : ЧОРО — ъ,чо : В,3ЧО - дуо гб,сю з 1000 Ив ; 26.72

Example 2: Collision of Two Moving Objects

  • On a snow-covered road, a car with a mass of 1100 kg collides head-on with a van having a mass of 2500 kg traveling at 8 m/s

  • As a result of the collision, the vehicles lock together and immediately come to rest.

  • Calculate the speed of the car immediately before the collision

    C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image133.png

Example 3: Recoil Velocity

  • A 4-kg rifle fires a 20-gram bullet with a velocity of 300 m/s. Find the recoil velocity of the rifle

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Example 4: Atomic Collision

  • A proton (mass=m) and a lithium nucleus (mass=7m) undergo an elastic collision as shown below.

  • Find the velocity of the lithium nucleus following the collision

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Example 5: Collisions in Multiple Dimensions

  • Bert strikes a cur ball of mass 0.17 kg , giving it a velocity of 3 m/s in the x-direction. When the cue ball strikes the eight ball (mass=0.16kg), previously at rest, the eight ball is deflected 45 degrees from the cur ball's previous path, and the cue ball is deflected 40 degrees in the opposite direction. Find the velocity of the cue ball and the eight ball after the collision

    3 m/s 0.17kg - 8=450 0.16kg

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2001 Free Response Question 1

Motion Sensor Mech 1. Force Sensor A motion sensor and a force sensor record the motion of a cart along a track, as shown above. The cart is given a push so that it moves toward the force sensor and then collides with it. The two sensors record the values shown in the following graphs. 0.30 0.20 - 0.10 - > -0.10 -0.20 0.30 0.38 0.40 50 40 - 30 - 20 10 0.30 0.32 0.34 0.36 Time t (s) 0.32 0.34 0.36 Time t (s) 0.38 (a) (b) (c) (d) Determine the cart's average acceleration between t = 0.33 s and t = 0.37 s. Determine the magnitude of the change in the cart's momentum during the collision. Determine the mass of the cart. Determine the energy lost in the collision between the force sensor and the cart.

C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image139.png

2002 Free Response Question 1

A crash test car of mass 1,000 kg moving at constant speed of 12 m/s collides completely inelastically with an object of mass M at time t = O. The object was initially at rest. The speed v in m/s of the car-object system after the collision is given as a function of time t in seconds by the expression (a) (b) (c) (d) 8 1+5t• Calculate the mass M of the object. Assuming an initial position of x = O, determine an expression for the position of the car-object system after the collision as a function of time t. Determine an expression for the resisting force on the car-object system after the collision as a function of time t. Determine the impulse delivered to the car-object system from t = O to t = 2.0 s.

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2014 Free Response Question 1

1 10 20 O Mech. 1. 30 2 40 50 Photogates 3 60 70 4 80 90 5 100 110 120 cm In an experiment, a student wishes to use a spring to accelerate a cart along a horizontal, level track. The spring is attached to the left end of the track, as shown in the figure above, and produces a nonlinear restoring force of magnitude FS = As + Bs, where s is the distance the spring is compressed, in meters. A measuring tape, marked in centimeters, is attached to the side of the track. The student places five photogates on the track at the locations shown. (a) Derive an expression for the potential energy U as a function of the compression s. Express your answer in terms of A, B, s, and fundamental constants, as appropriate.

In a preliminary experiment, the student pushes the cart of mass 0.30 kg into the spring, compressing the spring 0.040 m. For this spring, A = 200 N m and B = 150 N/m. The cart is released from rest. Assume friction and air resistance are negligible only during the short time interval when the spring is accelerating the cart. (b) Calculate the following: i. The speed of the cart immediately after it loses contact with the spring ii. The impulse given to the cart by the spring In a second experiment, the student collects data using the photogates. Each photogate measures the speed of the cart as it passes through the gate. The student calculates a spring compression that should give the cart a speed of 0.320 m/s after the cart loses contact with the spring. The student runs the experiment by pushing the cart into the spring, compressing the spring the calculated distance, and releasing the cart. The speeds are measured with a precision of ±0.002 m s. The positions are measured with a precision of ±0.005 m. Photogate Cart speed (m/s) Photogate position (m) 1 0.412 0.20 2 0.407 0.40 3 0.399 0.60 4 0.374 0.80 5 0.338 1.00 (c) On the axes below, plot the data points for the speed D of the cart as a function of position x. Clearly scale and label all axes, as appropriate.

(d) i. Compare the speed of the cart measured by photogate 1 to the predicted value of the speed of the cart just after it loses contact with the spring. List a physical source of error that could account for the difference. ii. From the measured speed values of the cart as it rolls down the track, give a physical explanation for any trend you observe.

a) —Falș u

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