A golf ball is hit off a tee at the edge of a cliff. Its x...

A golf ball is hit off a tee at the edge of a cliff. Its $x$ and $y$ coordinates versus time are given by the following expressions: $$ x=(18.0 \mathrm{~m} / \mathrm{s}) t $$ and $$ y=(4.00 \mathrm{~m} / \mathrm{s}) t-\left(4.90 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2} $$ (a) Write a vector expression for the ball's position as a function of time, using the unit vectors $\mathbf{i}$ and $\mathbf{j}$. By taking derivatives of your results, write expressions for (b) the velocity vector as a function of time and (c) the acceleration vector as a function of time. Now use unit vector notation to write expressions for (d) the position, (e) the velocity, and (f) the acceleration of the ball, all at $t=3.00 \mathrm{~s}$

A golf ball is hit off a tee at the edge of a cliff. Its $x$ and $y$ coordinates versus time are given by the following expressions:
$$
x=(18.0 \mathrm{~m} / \mathrm{s}) t
$$
and
$$
y=(4.00 \mathrm{~m} / \mathrm{s}) t-\left(4.90 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}
$$
(a) Write a vector expression for the ball's position as a function of time, using the unit vectors $\mathbf{i}$ and $\mathbf{j}$. By taking derivatives of your results, write expressions for (b) the velocity vector as a function of time and (c) the acceleration vector as a function of time. Now use unit vector notation to write expressions for (d) the position, (e) the velocity, and (f) the acceleration of the ball, all at $t=3.00 \mathrm{~s}$

Key Concepts

Acceleration as the First Derivative of Velocity

Acceleration is found by differentiating the velocity vector with respect to time. It represents the rate of change of velocity and is also expressed as a vector. In many kinematics problems, such as those involving projectile motion, the acceleration due to gravity is a constant that acts in the downward direction.

Velocity as the First Derivative of Position

The velocity vector of an object is obtained by differentiating its position vector with respect to time. This derivative yields the rate of change of the object’s position, and it is expressed as a vector that includes both magnitude and direction, reflecting the object's speed and direction in each dimension.

Projectile Motion Under Gravity

Projectile motion concerns the movement of objects that are launched into the air and subject to the acceleration due to gravity. In such scenarios, the horizontal component of motion is constant (if air resistance is neglected), while the vertical component is influenced by a uniform gravitational acceleration. This separation of motion into independent components is a key approach in solving these types of problems.

Unit Vector Notation

Unit vector notation is a method of representing vectors by defining their components in specific directions. Typically, the horizontal direction is represented by the unit vector i and the vertical direction by the unit vector j. These unit vectors have a magnitude of one and serve as the basis for expressing any vector in two-dimensional space.

Kinematic Equations in Vector Form

Kinematic equations describe the motion of an object and, when written in vector form, provide a concise way to represent position, velocity, and acceleration as functions of time. This framework allows for the independent analysis of horizontal and vertical components, particularly useful in problems involving projectile motion.

Position Vector in Two Dimensions

This concept involves representing the location of an object using a vector that includes both horizontal and vertical components. In physics, this is commonly expressed as a sum of the unit vectors i (for the x-axis) and j (for the y-axis), which together locate the object in a two-dimensional coordinate system.

Transcript

00:01 Hello.

00:02 In this question, we have a golf ball that has been hit from a cliff.

00:07 And this hit, of course, gave the golf ball an initial acceleration or an initial velocity to be accurate.

00:16 And then it starts to experience free fall.

00:20 This is not the main interest of the question, because here we're dealing with vector quantities.

00:25 We're giving that the x and y as functions of top.

00:33 Where x represents the position of the ball in the x direction and y represents the position of the ball in the y direction so here the x is basically 18 t and y is 4t minus 4 .9 t squared first we're asked to represent the position vector in the i and j unit vector notation so that's pretty straightforward because we know that r is basically the x in the direction of i plus y in the direction of j substituting this we'll have that the position is 18 t in the i direction and 4 t minus 4 .9 t squared in the j direction so that's the answer for part now let's move to a part b.

01:47 It's asking us by taking derivatives of this position vector just we just got here to represent that velocity of the golf ball as in vector notation as well in the i and j unit vectors.

02:02 So that's pretty straightforward.

02:04 We know that v is not delta because that's derivation here.

02:09 So that's the r by dt and differentiating this with respect to t, this will give us 18 in the i direction, and here we have 4 minus 2 multiplied by 4 .9, that's 9 .8 t in the direction of j, and that's the answer for part b.

02:40 In the third part or part c, by the same step that we did, which is just getting the derivative, we're asked to get the acceleration vector.

02:57 We know that the acceleration vector is just the derivative of the velocity vector with respect to time.

03:03 In doing this, we'll just give us zero in the eye direction, and this one will give us negative 9 .8 in the j direction.

03:14 So that's the answer for part c, but there is a note that i would like to stress on here.

03:23 Now you can see the number 9 .8, which is totally expected for a free fall.

03:29 And here the negative sign is to express that the acceleration is pointing downward towards the earth, assuming that the positive y axis, of course, is pointing upward to the sky...

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