Select the letter to the left of the applet to initialize Problem A.

- Shoot the bullet. It misses the target, because the initial speed is too low. Using DVAT equations, determine the value of the initial speed required to hit the target. Enter the calculated speed and test your prediction.
- Now change the target position and recalculate the initial speed. Test your prediction.
- How can you quickly determine the initial speed for any target position at the same gun height?
- Change the gravitational field strength to the value at the surface of the Moon. Predict the initial speed required to hit the target. Test your prediction.

Select the letter to the left of the applet to initialize Problem B.

- A bullet will be shot from ground level at 45°. Using DVAT equations, calculate the following:

- initial speed needed to hit the target,
- time that the bullet is in flight,
- impact speed of the bullet.

- Test your predictions.

Select the letter to the left of the applet to initialize Problem C.

- Ignoring the target position, shoot the bullet now.
- Pause the motion when the bullet strikes the ground. Use the step frame buttons (>> or <<) as needed to position the ball precisely at ground level.
- Note the x-coordinate of the bullet and enter this as the Target Position.
- Shoot the bullet to verify that you hit the target this time.
- Determine a second angle for which the bullet hits the target in the same position.
- Change the angle to 25° and repeat steps 1-5.
- Now select 15° and repeat steps 1-5 again. Do you see a pattern emerging? What is it?

Select the letter to the left of the applet to initialize Problem D.

- A bullet will be shot from ground level at 20°. Using DVAT equations, calculate the following:

- initial speed needed to hit the target,
- time that the bullet is in flight,
- impact speed of the bullet.

- Test your predictions. Then predict a different angle that will allow the projectile to hit the target at the same position.

Select the letter to the left of the applet to initialize Problem E.

- A projectile is shot from a cliff at a 45° angle. Determine the initial speed of the bullet required to hit the target.
- Enter the calculated speed and test your prediction.

Select the letter to the left of the applet to initialize Problem F.

- A projectile is fired at a speed of 50 m/s from a gun on a cliff. Using DVAT equations, determine an angle for which the bullet hits the target. Test your prediction.
- Determine a second angle for which the bullet hits the target, given the same initial speed. Test your prediction.

Select the letter to the left of the applet to initialize Problem G.

- In this problem, the target is at the origin when the projectile is launched. The target has a constant velocity of 25 m/s to the right. If the projectile is launched horizontally, what must its initial speed be in order to hit the target? Explain your answer and then test it.
- If the height of the cliff is 50% greater, what will the initial speed have to be in order to hit the moving target? Explain your prediction and then test it.
- If the projectile is launched on the Moon, what will the initial speed have to be? Again, explain and test.
- Suppose now that the projectile is launched at a 60° angle on the Earth. What will the initial speed of the projectile have to be in order to hit the target? Explain and test.

Select the letter to the left of the applet to initialize Problem H.

- In this problem, determine the constant velocity that the target must have in order to intercept the projectile.
- If the target moves to the left with the speed determined in question 1, what must its initial position be?

Select the letter to the left of the applet to initialize Problem I.

- The target starts moving from rest from the origin at the same time that the projectile is launched horizontally. Determine:

- the acceleration of the target required to intercept the projectile,
- where the intercept occurs,
- the velocity of the target at intercept. Test your predictions.

- Repeat problem 1 for each of the following initial target velocities. (Find a way to simplify the number of calculations you must do.) a. 10 m/s b. 40 m/s c. 60 m/s d. 80 m/s e. 160 m/s.

Select the letter to the left of the applet to initialize Problem J.

- A projectile is launched from ground level at 45°. At the same instant, the target--initially at the origin--accelerates uniformly from rest. Determine the acceleration of the target required to intercept the projectile.
- If the initial speed of the projectile is 30 m/s but the angle of launch remains the same, what must the acceleration of the target be?
- If the launch takes place on the Moon, what must the acceleration of the target be? (Select an initial projectile speed so that the intercept point is on screen.)
- By now, you should have seen a pattern. You should have also obtained a single formula that relates three things: a) angle of launch, b) gravitational field strength, c) acceleration of target. Using this relationship, determine the angle of launch on planet X (g=20 N/kg) for a target acceleration of 30 m/sē. Oh, by the way, select the initial projectile speed so that the range is 375 m.

Select the letter to the left of the applet to initialize Problem K.

The following problems are borrowed from "Solving an 'Unsolvable' Projectile-Motion Problem", David Montalvo, TPT, vol.37, April, '99, p.226. The 3 problems are variations on the same situation. Sure, they're easy to do by trial and error. But can you successfully predict the answers before testing them? An explicit solution in terms of q isn't possible. However, a graphing calculator may come in handy.The Gunner's Problem: A cannon fires a projectile in order to intercept a tank moving toward the cannon's position at constant velocity. Without changing the speed of the projectile or the velocity of the tank, find the angle of the cannon necessary to strike the target. (There are 2 solutions.)

Reinitialize Problem K to continue.

The Yeoman's Problem: An arrow is shot from a bow at a fixed angle. Find the speed of the arrow needed to strike a target moving toward the bow at constant velocity.

Reinitialize Problem K to continue.

The Wide Receiver's Problem: A quarterback throws a football at a given angle and speed. Find the velocity at which the receiver must run to catch the ball.