# Calculate Initial Acceleration Needed to Throw a Ball from Point A to B

• Bosduif
In summary, the conversation discusses how to calculate the initial acceleration needed for a ball to land on a specific point when thrown at a 45 degree angle. The conversation also clarifies the use of velocity and acceleration and how they relate to position and time. The problem is simplified to a two-dimensional scenario and the solution is presented through defining and using velocity and acceleration vectors.

Hello

## Homework Statement

I have two known points A and B. I want to throw a ball from A to B with a certain angle. Obviously I also know the gravity.

For example: (in metres)

Known
point A: (0, 0, 0)
point B: (30, 0, 10)
direction: (0.949, 0, 0.316)
gravity: 9.82 m/s
angle: 45° or PI / 4

Wanted
acceleration: ?

How can I calculate the initial acceleration needed at point A for the ball to land on point B?

Do you mean initial velocity?

If you do mean initial acceleration, then how is that acceleration assumed to be applied to the ball, i.e., is it constant acceleration for some time window, or what?

Yes, I meant velocity, the initial speed at which the ball is thrown
I'm sorry, English isn't my primary language and I always get the two mixed up

Bosduif said:
Yes, I meant velocity, the initial speed at which the ball is thrown
I'm sorry, English isn't my primary language and I always get the two mixed up

No problem.

I would like to clarify one more thing. Are the (x,y,z) coordinates oriented such that +z = up, i.e. point B is 10 meters higher than point A?

Oh, I'm using a Y is up system but the position doesn't really matter.
The final purpose is that I have a thrower and a target which moves around. The thrower throws a ball with a certain speed towards the target. The ball always has to hit the target.

In one case know the starting point, the point of destination and the angle (direction vector) and I need to know the speed to multiple with the direction so that it will hit the target.

Bosduif said:
Oh, I'm using a Y is up system but the position doesn't really matter.

Well, the vertical coordinate does matter. If the points are at different heights then certain throwing angles might not even work, no matter what velocity you use.

But let's assume both points are at the same height, and we don't even need to work in three dimensions. Just use a coordinate that points from A to B. All that matters is their distance apart. Furthermore, we can define the origin to be point A.

So let's reformulate the problem as follows:

I want to throw a ball from the origin (0,0) to some point (d,0), where d is a horizontal distance. The second (y) component is height.

The angle is constrained to be 45 degrees. Choose an initial velocity so that the ball will hit the target.

We can define the velocity as a function of time as follows.

$$v(t) = [v_x(t), v_y(t)]$$

where $v_x(t)$ is the horizontal velocity and $v_y(t)$ is the vertical velocity. We will assume the ball is thrown at $t = 0$.

Note that the 45 degree throwing angle means

$$v_x(0) = v_y(0)$$

We can also define the position as a function of time:

$$p(t) = [p_x(t), p_y(t)]$$

where we assume

$$p_x(0) = p_y(0) = 0$$

The velocity and position vectors are related as follows

$$\frac{d}{dt}p(t) = v(t)$$

Finally, we can define an acceleration vector

$$a(t) = [a_x(t), a_y(t)]$$

which relates to velocity as follows

$$\frac{d}{dt} v(t) = a(t)$$

So let's start with acceleration. What are $a_x(t)$ and $a_y(t)$?

## 1. How do you calculate the initial acceleration needed to throw a ball from point A to B?

To calculate the initial acceleration needed to throw a ball from point A to B, you would need to use the formula: a = (vf - vi)/t, where a is the acceleration, vf is the final velocity, vi is the initial velocity, and t is the time taken. You would also need to consider factors such as the angle of the throw, air resistance, and gravity.

## 2. What is the significance of calculating the initial acceleration for throwing a ball?

Calculating the initial acceleration for throwing a ball is important because it helps determine the amount of force needed to throw the ball accurately and reach a specific point. It also allows for adjusting the throw to account for external factors such as wind or air resistance.

## 3. How does the angle of the throw affect the initial acceleration needed?

The angle of the throw can significantly affect the initial acceleration needed to throw a ball from point A to B. Throwing at a higher angle will require a greater initial acceleration to overcome the force of gravity and reach the same point, while throwing at a lower angle will require less initial acceleration.

## 4. How does air resistance impact the calculation of initial acceleration?

Air resistance can affect the initial acceleration needed to throw a ball by reducing the ball's speed and distance traveled. This is because air resistance creates a force in the opposite direction of the ball's motion, which must be overcome by a greater initial acceleration to reach the desired point.

## 5. Can you calculate the initial acceleration needed for any object, not just a ball?

Yes, the formula for calculating initial acceleration can be applied to any object, as long as the necessary variables are known. However, the impact of external factors such as air resistance may vary for different objects and may need to be taken into account in the calculation.