## Sl

Fig 22.16.

Analogy for gravity assist. (a) A low mass ball with speed v approaches a high mass train with speed u for a head-on collision. (b) The same situation viewed from the train.The train is at rest, and the ball has a speed v + u. (c) Still looking from the train, the elastic collision simply reverses the direction of the ball, so it is now moving away from the train at a speed v + u. (d) We revert to the view from the ground. If the ball is moving ahead of the train at a speed v + u, then the speed of the ball, relative to the ground, must be this speed, plus the speed of the train u, giving a speed v + 2u.

is much less than the mass of the train, and that the collision is completely elastic. The train is moving with speed u and the ball is moving with speed v. As viewed from the train, the ball is coming towards the train at a speed of u + v. In the elastic collision, the train is so massive that it doesn't recoil, so the ball simply reverses its velocity relative to the train. It is now moving away from the train at a speed u + v. As viewed from the ground, its speed is now v + 2u. In the collision, the ball has taken a little energy from the train (very small relative to the total energy of the train), and its speed is now greater than its initial speed by twice the speed of the train.

With a space probe, We cannot, of course, cause a space probe to bounce directly off Jupiter.

Actual Trajectory window. The size of the window depends on how far we are willing to deviate from the minimum energy orbit. In particular, orbits with higher energy might be used since they have shorter travel times. Since the launch requires the correct relative positions of the Earth and Venus, the launch window opens once per synodic period of Venus. This explains the spacing of launches for a given planet.

There is a trick that can be used to minimize the energy needed to visit a planet beyond Jupiter. In this, we take advantage of an elastic gravitational encounter between the space probe and a massive planet. This is known as a gravity assist.

To understand how this works, let's look at an analogous situation, a ball bounced off the front of an incoming train. This is illustrated in Fig. 22.16. We assume that the mass of the ball

Actual Trajectory Trajectory that would give maximum boost

Trajectory that would give maximum boost

Spacecraft

### Fig 22.17.

Trajectories for gravity assist.The space probe doesn't have to collide with the planet; it only has to make a close gravitational encounter.The maximum boost would come from a trajectory that comes the closest to a head-on collision. However, since the spacecraft is coming from well inside the planet's orbit, this is not practical.The actual trajectory still gives almost half of the maximum boost in speed.

However, we can arrange the trajectory to have the same effect (Fig. 22.17). We cannot generally take advantage of a head-on collision, but the speed of the probe can be increased by an amount of the order of the speed of Jupiter. The technique has been used for the Voyager 1 and 2 missions, as shown in Fig. 22.18. Fig 22.18.

Orbit of Voyager 2 showing the effect of gravitational assist in traveling past each planet.

Fig 22.18.

Orbit of Voyager 2 showing the effect of gravitational assist in traveling past each planet.