Principles of Mechanics pp 7385  Cite as
Impulse, Momentum, and Collisions
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Abstract
When two billiard balls collide, in which direction would they travel after the collision? If a meteorite hits the earth, why does the earth remain in its orbit? When two cars collide with each other, why is one of the cars more damaged than the other? We will find that to answer such questions, new concepts must be introduced.
5.1 Linear Momentum and Collisions
When two billiard balls collide, in which direction would they travel after the collision? If a meteorite hits the earth, why does the earth remain in its orbit? When two cars collide with each other, why is one of the cars more damaged than the other? We will find that to answer such questions, new concepts must be introduced.
Consider the situation where two bodies collide with each other. During the collision, each body exerts a force on the other. This force is called an impulsive force, because it acts for a short period of time compared to the whole motion of the objects, and its value is usually large. To solve collision problems by using Newton’s second law, it is required to know the exact form of the impulsive forces. Because these forces are complex functions of the collision time, it is difficult to find their exact form and would make it difficult to use Newton’s second law to solve such problems. Thus, new concepts known as momentum and impulse were introduced. These concepts enable us to analyze problems that involve collisions, as well as many other problems.
5.2 Conservation of Linear Momentum
5.3 Impulse and Momentum
5.4 Collisions
As discussed previously, when two bodies collide, they exert large forces on one another (during the time of the collision) called impulsive forces. These forces are very large such that any other forces ( \(\mathrm {e}.\mathrm {g}\)., friction or gravity) present during the short time of the collision can be neglected. This approximation is known as the impulse approximation. For example, if a golf ball was hit by a golf club, the change in the momentum of the ball can be assumed to be only due to the impulsive force exerted on it by the club. The change in its momentum due to any other force present during the collision can be neglected. That is, the force in the expression I \(=\triangle \mathrm {p}=\overline{\mathrm {F}}\triangle t\) can be assumed to be the impulsive force only The neglected forces present during the collision time are external to the twobody system, whereas the impulsive forces are internal. The twobody system can therefore be considered to be isolated during the short time of the collision (which is in the order of a few milliseconds). Hence, the total linear momentum of the system is conserved during the collision, which enables us to apply the law of conservation of momentum immediately before and immediately after the collision. In general, for any type of collision, the total linear momentum is conserved during the time of the collision. That is, \(\mathrm {p}_{i}=\mathrm {p}_{f}\)., where \(\mathrm {p}_{i}\) and \(\mathrm {p}_{f}\) are the momenta immediately before and after the collision. In the next sections, we will define various types of two body collisions, depending on whether or not the kinetic energy of the system is conserved.
Example 5.1
A 50 \(\mathrm {g}\) golf ball initially at rest is struck by a golf club. The golf club exerts a force on the ball that varies during a very short time interval from zero before impact, to a maximum value and back to zero when the ball is no longer in contact with the club. If the ball is given a speed of 25 \(\mathrm {m}/\mathrm {s}\), and if the club is in contact with the ball for \(7\times 10^{}4\,\mathrm {s}\), find the average force exerted by the club on the ball.
Solution 5.1
Example 5.2
A canon placed on a carriage fires a 250 kg ball to the horizontal with a speed of 50 \(\mathrm {m}/\mathrm {s}\). If the mass of the canon and the carriage is 4000 kg, find the recoil speed of the canon.
Solution 5.2
Example 5.3
A hockey puck of mass 0.16 kg traveling on a smooth ice surface collides with the court’s edge. If its initial and final velocities are \(\mathbf {v_{i}}=2 \; \mathbf {i}\mathrm {m}/\mathrm {s}\) and \(\mathbf {v_{f}}=1 \; \mathbf {i}\mathrm {m}/\mathrm {s}\) and if the hockey puck is in contact with the wall for 2 ms, find the impulse delivered to the puck and the average force exerted on it by the wall.
Solution 5.3
Example 5.4
A 0.5 kg hockey puck is initially moving in the negative \(\mathrm {y}\)direction as shown in Fig. 5.3, with a speed of 7 \(\mathrm {m}/\mathrm {s}\). If a hockey player hits the puck giving it a velocity of magnitude 12 \(\mathrm {m}/\mathrm {s}\) in a direction of \(60^{\mathrm {o}}\) to the vertical, and if the collision lasts for 0.008 \(\mathrm {s}\), find the impulse due to the collision and the average force exerted on the puck.
Solution 5.4
Example 5.5
Two ice skaters of masses \(m_{1}=50\) kg and \(m_{1}=62\) kg standing face to face push each other on a frictionless horizontal surface. If skater (1) recoils with a speed of 5 \(\mathrm {m}/\mathrm {s}\), find the recoil speed of the other skater.
Solution 5.5
Example 5.6
Solution 5.6
5.4.1 Elastic Collisions
An elastic collision is one in which the total kinetic energy, as well as momentum, of the twocollidingbody system is conserved. These collisions exist when the impulsive force exerted by one body on the other is conservative. Such force converts the kinetic energy of the body into elastic potential energy when the two bodies are in contact. It then reconverts the elastic potential energy into kinetic energy when there is no more contact. After collision, each body may have a different velocity and therefore a different kinetic energy. However, the total energy as well as the total momentum of the system is constant during the time of the collision. An example of such collisions is those between billiard balls.
5.4.2 Inelastic Collisions
5.4.3 Elastic Collision in One Dimension
5.4.3.1 Special Cases
1. If \(m_{1}=m_{2}\), it follows from Eqs. 5.2 and 5.3 that \(v_{1f}=v_{2i}\) and \(v_{2f}= v_{1i}\). In other words, if the particles have equal masses they exchange velocities.
5.4.4 Inelastic Collision in One Dimension
5.4.5 Coefficient of Restitution

If \(e=1\) the collision is perfectly elastic.

If \(e<1\) the collision is inelastic.

If \(e=0\) the collision is perfectly inelastic (the two bodies stick together).
Example 5.7
Two marble balls of masses \(m_{1}=7\) kg and \(m_{2}=3\) kg are sliding toward each other on a straight frictionless track. If they experience a headon elastic collision and if the initial velocities of \(m_{1}\) and \(m_{2}\) are 0.5 \(\mathrm {m}/\mathrm {s}\) to the right and 2 \(\mathrm {m}/\mathrm {s}\) to the left, respectively, find the final velocities of \(m_{1}\) and \(m_{2}.\)
Solution 5.7
Example 5.8
The ballistic pendulum consists of a large wooden block suspended by a light wire (see Fig. 5.6). The system is used to measure the speed of a bullet where the bullet is fired horizontally into the block. The collision is perfectly inelastic and the system (bullet\(+\)block) swings up a height h. If \(M=3\) kg, \(m=5\,\mathrm {g}\) and \(h=5\) cm, find (a) the initial speed of the bullet; (b) the mechanical energy lost due to the collision.
Solution 5.8
Example 5.9
Two masses \(m_{1}=0.8\) kg and \(m_{2}=0.5\) kg are heading toward each other with speeds of 0.25 \(\mathrm {m}/\mathrm {s}\) and \(0.5\,\mathrm {m}/\mathrm {s}\), respectively. If they have a perfectly inelastic collision, find the final velocity of the system just after the collision.
Solution 5.9
Example 5.10
Two blocks \(m_{1}=2\) kg and \(m_{2}=1\) kg collide headon with each other on a frictionless surface (see Fig. 5.7. If \(v_{1i}=10\,\mathrm {m}/\mathrm {s}\) and \(v_{2i}=15\,\mathrm {m}/\mathrm {s}\) and the coefficient of restitution is \(e=1/4\), determine the final velocities of the masses just after the collision.
Solution 5.10
Example 5.11
A \(m_{1}=5\,\mathrm {g}\) bullet is fired horizontally at the center of a wooden block with a mass of \(m_{2}=2\,\mathrm {k}\mathrm {g}\). The bullet embeds itself in the block and the two slides a distance of 0.\(5\,\mathrm {m}\) on a rough surface \((\mu _{k}=0.2)\) before coming to rest. Find the initial speed of the bullet.
Solution 5.11
5.4.6 Collision in Two Dimension
Example 5.12
A ball of mass of 2 kg is sliding along a horizontal frictionless surface at a speed of 3 \(\mathrm {m}/\mathrm {s}\). It then collides with a second ball of mass of 5 kg that is initially at rest. After the collision, the second ball is deflected with a speed of 1 \(\mathrm {m}/\mathrm {s}\) at an angle of \(30^{\mathrm {o}}\) below the horizontal as shown in Fig. 5.9. (a) Find the final velocity of the first ball; (b) show that the collision is inelastic; (c) suppose that the two balls have equal masses and the collision is perfectly elastic, show that \(\theta _{1}+\theta _{2}=90^{\mathrm {o}}.\)
Solution 5.12
Example 5.13
A 1200 kg car traveling east at a speed of 18 \(\mathrm {m}/\mathrm {s}\) collides with another car of mass of 2500 kg that is traveling north at a speed of 23 \(\mathrm {m}/\mathrm {s}\) as shown in Fig. 5.10. If the collision is perfectly inelastic, how much mechanical energy is lost due to the collision?
Solution 5.13
5.5 Torque
Example 5.14
A force \(\mathbf {F}=(2t\mathbf {i}(t^{2}3)\mathbf {j}+4t^{5}\mathbf {k}) \; \mathrm {N}\) acts on a particle that has a position vector \(\displaystyle \mathbf {r}=\bigg (6\mathbf {i}+5t\mathbf {j}+(\frac{t}{2}1) \; \mathbf {k}\bigg )\,\mathrm {m}\) find the torque of the particle about the origin at \(t=1\,\mathrm {s}.\)
Solution 5.14
5.6 Angular Momentum
5.6.1 Newton’s Second Law in Angular Form
5.6.2 Conservation of Angular Momentum
Example 5.15
A cat watches a mouse of mass m run by, as shown in Fig. 5.14. Determine the mouse’s angular momentum relative to the cat as a function of time if the mouse has a constant acceleration a and if it starts from rest.
Solution 5.15
Example 5.16
A 0.2 kg particle is moving in the x–y plane. If at a certain instant \(r=3\,\mathrm {m}\) and \(v=10\,\mathrm {m}/\mathrm {s}\) (see Fig. 5.15), find the magnitude and direction of the angular momentum of the particle at that instant relative to the origin.
Solution 5.16
Example 5.17
A particle is moving under the influence of a force given by \(\mathbf {F}=k\mathbf {r}\). Prove that the angular momentum of the particle is conserved.
Solution 5.17
Example 5.18
A particle is moving in a circle where its position as a function of time is given by the expression \(\mathbf {r}=a(\cos \omega t\mathbf {i}+\sin \omega t\mathbf {j})\) , where \(\omega \) is a constant. Show that the total angular momentum of the particle is constant.
Solution 5.18
 1.
A tennis ball of mass of 0.06 kg is initially traveling at an angle of \(47^{\mathrm {o}}\) to the horizontal at a speed of 45 \(\mathrm {m}/\mathrm {s}\). It then was shot by the tennis player and return horizontally at a speed of 35 \(\mathrm {m}/\mathrm {s}\). Find the impulse delivered to the ball.
 2.
A force on a 0.5 kg particle varies with time according to Fig. 5.16. Find (a) The impulse delivered to the particle, (b) the average force exerted on the particle from \(t=0\) to \(t=6\,\mathrm {s}(\mathrm {c})\). The final velocity of the particle if its initial velocity is 2 \(\mathrm {m}/\mathrm {s}.\)
 3.
A 1 kg particle moves in a force field given by \(\mathbf {F}=(2t^{2}\mathbf {i}+(5t3)\mathbf {j}6t\mathbf {k})\) N. Find the impulse delivered to the particle during the time interval from \(t=1\,\mathrm {s}\) to \(t=3\,\mathrm {s}.\)
 4.
A boy of mass 45 kg runs and jump with a horizontal speed of 4.5 \(\mathrm {m}/\mathrm {s}\) into a 70 kg cart that is initially at rest (see Fig. 5.17). Find the final velocity of the boy and the cart.
 5.
A rubber ball of mass of 0.2 kg is dropped from a height of 2.2 \(\mathrm {m}\). It re bounds to a height of 1.1 \(\mathrm {m}\). Find (a) the coefficient of restitution, (b) the energy lost due to impact.
 6.
A 1200 kg car initially traveling at 12 \(\mathrm {m}/\mathrm {s}\) due east collides with another car of mass of 1600 kg that is initially at rest. If the cars become entangled after the collision, find the common final speed of the cars.
 7.
Figure 5.18 shows a ball that strikes a smooth surface with a velocity of 20 \(\mathrm {m}/\mathrm {s}\) at an angle of \(45^{\mathrm {o}}\) with the horizontal. If the coefficient of restitution for the impact between the ball and the surface is \(e=0.85\), find the magnitude and direction of the velocity in which the ball rebounds from the surface. (Hint: use the velocity components in the direction perpendicular to the surface for the coefficient of restitution).
 8.
Two gliders moving on a frictionless linear air track experience a perfectly elastic collision (see Fig. 5.19). Find the velocity of each glider after the collision.
 9.
A bullet of mass of m is fired with a horizontal velocity v into a block of mass M. The block is initially at rest on a frictionless surface and is connected to a spring of force constant of k (see Fig. 5.20). If the bullet embeds itself in the block causing the spring to compress to a maximum distance d, find the initial speed of the bullet.
 10.
A block moves along the \(\mathrm {y}\)axis due to a force given by \(\mathbf {F}=a\mathbf {i}\) (see Fig. 5.21). Find the torque on the block about (a) the origin (b) point A.
 11.
A conical pendulum of mass m and length L is in uniform circular motion with a velocity v (see Fig. 5.22). Find the angular momentum and torque on the mass about O.
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