Example Question #11 : Newtonian Mechanics
If a 15kg ball takes five seconds to strike the ground when released from rest, at what height was the ball dropped?
Using the equation we can find the distance at which the ball was dropped. Notice that the mass of the ball does not matter in this problem. We are told that the ball is dropped from rest making, , thus we have . When we plug in our values, and assuming that acceleration is equal to gravity (10m/s2) we find that = 125m.
Example Question #1 : Calculating Motion In One Dimension
How far will an object travel after ten seconds if it is dropped into a bottomless pit?
Since the object is dropped, the inital velocity is zero. Gravity is the only acceleration, the time is ten seconds, and the distance at which the object travels is unknown.
The equation can be used to find the distance traveled.
Example Question #2 : Calculating Motion In One Dimension
How long does it take an object to travel a distance of 30m from rest at a constant acceleration of 2m/s2?
Using the equation , we can solve for time.
Since the object started at rest, . Now we are left with the equation .
Plugging in the remaining values we can find that t = 5.5s.
Example Question #3 : Motion In One Dimension
A person on top of a tall building drops a rock. How long will it take for the rock to reach the ground? Ignore air resistance.
Use the following kinematic equation: .
We can choose the ground to be the zero distance so that and .
Also, the initial speed is zero.
The kinematic equation simplifies using these values.
Solve to isolate the time variable.
We know that the acceleration is the acceleration due to gravity. Now we can plug in the known values and solve.
Example Question #13 : Newtonian Mechanics
You are driving at a speed of and suddenly, a tree falls down on the road blocking your path. You slam on your brakes to avoid hitting the fallen tree and thus, come to a complete stop. You were at a distance of away from the tree when you hit the brakes. Assuming that your vehicle does not skid, what is the minimum deceleration needed to avoid hitting the fallen tree?
Use the following kinematic equation where the initial velocity is , final velocity is , and the distance traveled is .
We can use the values in the question to solve for the acceleration.
We rearrange the equation to solve for the acceleration.
Example Question #4 : Motion In One Dimension
A is dropped from a height of . A picture is taken when the ball is from the ground with an exposure time of . If the actual diameter of the ball is , what will the vertical diameter of the ball appear to be in the picture?
The first step to solving this problem will be to find the velocity of the ball at the point when the picture is taken. We know the initial velocity of the ball (zero), the displacement, and the acceleration. Using the appropriate kinematics formula, we can solve for the final velocity.
Now that we know how fast the ball is traveling when the picture is taken, we can find the distance it travels while the shutter is open. This distance will become a motion blur, making the vertical diameter of the ball appear stretched.
During the exposure period, the ball will travel , or . The diameter of the ball in the vertical direction will appear to be distorted by this distance.
Example Question #5 : Linear Motion And Momentum
A person stands on the edge of a straight -high cliff and holds a ball over the edge. The person tosses the ball directly upward with an initial speed of . How long will it take the ball to hit the ground at the base of the cliff, below?
Assume for gravitational acceleration.
This is projectile motion in the vertical direction only, subject to the equation of motion: .
For this discussion, one can define the downward direction as negative. For projectile motion, (gravitational acceleration, or ).
In this case, the ball ends up below where is started, so .
The initial velocity, , is (upward, thus positive).
With all this, the projectile motion equation becomes:
This can be solved for using the quadratic formula:
The result is:
Only the positive answer option is physically possible, and is thus our correct answer.
Example Question #3 : Calculating Motion In One Dimension
A block is placed at a height of on a ramp at an incline of . It slides down from rest with an acceleration of due to a frictional force. What is the velocity of the block when it reaches the bottom of incline?
To find the final velocity we use the following kinematic equation.
The block starts from rest, so . The incline is eighteen meters high, so we can find the distance (hypotenuse) as follows:
We know that the acceleration is .
Plug in the values to find the velocity at the bottom of the incline.