Circular motion with constant acceleration problem

This problem from Serway and Jewett’s Physics for scientists and engineers with modern physics textbook (9th edition)

problem-42-circular-motion

Let’s analyze the wording of this problem:

  1. A ball swings in a vertical circle… then it is a circular motion problem.
  2. A given acceleration at specific position in space which is 36.9° past the lowest point on way up for the ball..then the given acceleration does not seem not to be a radial acceleration (see the diagram below in solution).

The information given in this problem:

  1. Radius of circular motion is 1.5 m (R=1.5m).
  2. The acceleration of the ball is a=-22.5iˆ+20.2jˆ m/s² at 36.9° away from the vertical axis (y-axis).

What the problem need:

  1. Sketch the vector diagram showing the components of the acceleration.
  2. Find the magnitude of radial acceleration.
  3. Find the speed and velocity of the ball.

 

SOLUTION:

  1. Sketch the vector diagram showing the components of the acceleration.

The diagram of acceleration vector

problem-42-circular-motion-a

2. Find the magnitude of radial acceleration.

The magnitude of radial acceleration could be found by using the horizontal and vertical components of the given acceleration:

problem-42-circular-motion-b

3. Find the speed and velocity of the ball.

The speed could be found using the radial acceleration and the radius of the circular motion:

problem 42 circular motion (c).jpg

Projectile Problem

Problem from Young and Freedman’s University Physics with Modern Physics textbook

problem-3-59-projectile

First we need to analyze the wording of the problem:

  1. A snow ball rolls off a barn roof.. that mean it is a projectile problem
  2. The height is y1=14 m and y2=0 m (the ground) then Δy=-14 m
  3. The range is x2= 4 m and x1=0 m (the barn side has the x-axis origin).

What are the given information:

  1. Angle at which the ball is leaving the roof is 40°.
  2. The initial velocity (vo) =7 m/s
  3. Δy=-14 m
  4. Δx=4 m

What the problem need:

  1. How far the ball will fall away from the barn (Δx=?)
  2. Draw x-t, y-t, vx-t, and vy-t graphs for the motion in part (1)
  3. If a man (his height= 1.9m) is standing at Δx=4 m, do the ball hit him when it falls?

SOLUTION:

  1. How far the ball will fall away from the barn (Δx=?)

problem-3-59-projectile-v

problem-3-59-projectile-a

2. Draw x-t, y-t, vx-t, and vy-t graphs for the motion in part (1)

problem-3-59-projectile-b

3. If a man (his height= 1.9m) is standing at Δx=4 m, do the ball hit him when it falls?

For this question, it is easy to find the time when Δx=4 m:

problem-3-59-projectile-c

While the man height is 1.9 m then the ball will be far from hitting him and he will be fine.

 

Projectile Problem

The following problem is from Young and Freedman’s University Physics with modern physics:

problem-3-39-projectile

Let’s analyze some of the wording of the problem:

  1. A rocket is fired at an angle from the top of a tower…that means it is a projectile problem.
  2. The height of tower ho= 49.3 m and the origin of coordinates is at the base of the tower.. then Δy=y2-y1=0-49.3= -49.3 m

Now what are the given information:

  1. h0= 49.3 m or we can say Δy= -49.3 m
  2. Position coordinates are x(t) and y(t) given by x(t)=A+Bt2 and y(t)=C+Dt3. We can say that at t=0 s, x(0)=A and y(0)=C. When it is a tower and the base of the tower is the origin of the coordinates then x(0)=0 m and y(0)=49.3 m.
  3. Acceleration of the rocket at time t= 1.5 s is a=5.4i+3.2j m/s2

What the problem needs:

  1. What are A,B,C, and D constants?
  2. What are the acceleration vector (a=?) and velocity vector (v=?) at the moment it got fired?
  3. What are the x- and y- component of velocity at time t=17.1 s after firing the rocket?
  4. What are the x- and y-component of position at time t=17.1s after firing the rocket?

SOLUTION:

  1. What are A,B,C, and D constants?

At first we can see that when x(0)=0 m = A then A=0 m and when y(0)=49.3 m = C then C= 49.3 m. We can use the the coordinate function x(t) and y(t) to get v(t) and a(t).

problem-3-39-projectile-sola

2. What are the acceleration vector (a=?) and velocity vector (v=?) at the moment it got fired?

Velocity and acceleration vectors already written above, then velocity and acceleration at the moment of firing the rocket are:

problem-3-39-projectile-solb

3. What are the x- and y- component of velocity at time t=17.1 s after firing the rocket?

problem-3-39-projectile-solc

4. What are the x- and y-component of position at time t=17.1s after firing the rocket?

problem-3-39-projectile-sold