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Chapter 3: Problem 49

\- A tiger leaps horizontally out of a tree that is \(4.00 \mathrm{~m}\) high. If he lands \(5.00 \mathrm{~m}\) from the base of the tree, calculate his initial speed. (Neglect any effects due to air resistance.) SSM

Short Answer

Expert verified
The initial speed of the tiger is approximately \(5.52 \mathrm{~m/s}\).

Step by step solution

01

Calculate time for vertical motion

First, calculate the time it takes for the tiger to fall down from the 4.00 m high tree. Assuming acceleration due to gravity to be \(9.81 m/s^2\) (downward), this can be done using the equation for distance in uniformly accelerated motion which is \(d = ut + 0.5*a*t^2\). Since the initial vertical velocity \(u\) is 0 (tiger jumps horizontally), the equation simplifies to \(d = 0.5*a*t^2\). Solving for \(t\) gives \(t = \sqrt{2d/a}\).
02

Substitute values to find time

Substitute the given height \(d = 4.00 m\) and \(a = 9.81 m/s^2\) into the formula. Hence, \(t = \sqrt{2*4.00/9.81}\). Calculating this gives approximately \(t \approx 0.905 s\).
03

Calculate horizontal speed

Now, calculate the initial horizontal speed of the tiger. The horizontal motion can be represented by the equation \(d = u*t\), where \(d\) is the distance the tiger lands from the base of the tree and \(u\) is the speed we want to find. Since there's no acceleration horizontally, we can solve for \(u\) to give \(u = d/t\).
04

Substitute values to find speed

Substitute \(d = 5.00 m\) and the time \(t = 0.905 s\) calculated previously into the speed formula to find the initial speed. Therefore, calculating \(u = 5.00/0.905\) gives \(u \approx 5.52 m/s\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Motion
When studying projectile motion, it's important to understand that horizontal and vertical motions are analyzed separately. Horizontal motion occurs when an object moves along a straight path on a horizontal plane.
In this case, horizontal motion is key to determining the speed at which a tiger leaps out of a tree.

The tiger lands 5 meters away from the tree's base. This distance is purely due to horizontal velocity, because no horizontal forces apply after the jump (neglecting air resistance).
  • We use the formula: \( d = u \times t \) where \(d\) is the horizontal distance (5 m), \(u\) is the initial speed, and \(t\) is the time taken.
  • Given, the horizontal motion has no acceleration, the velocity remains constant.
  • To calculate initial speed, rearrange the formula: \( u = \frac{d}{t} \).
From the example, the tiger's horizontal speed is 5.52 m/s.
This understanding is crucial for solving similar problems involving horizontal movements.
Vertical Motion
Vertical motion in projectile problems involves objects falling under the influence of gravity.
This is how we analyze the tiger's descent from the tree.

Gravity pulls the tiger down, affecting its vertical motion.
  • The formula used for vertical distance is: \( d = \frac{1}{2} a t^2 \), where \(a\) is gravitational acceleration (\(9.81 \text{ m/s}^2\)), \(t\) is the time, and the initial vertical velocity is 0.
  • We calculate how long it takes the tiger to hit the ground using \( t = \sqrt{\frac{2d}{a}} \), which relies on knowing the height of the fall.
The tiger falls 4 meters, so we calculate approximately 0.905 seconds for the fall.

Understanding vertical motion separately allows us to determine how long an object is in the air, which is crucial in combined motion problems.
Uniformly Accelerated Motion
Uniformly accelerated motion involves objects under a constant acceleration.
In most projectile motion problems, this happens due to gravity.

Here, the tiger falls with uniform acceleration due to Earth's gravity, as it leaps horizontally.
  • The key formula used is \( d = \frac{1}{2} a t^2 \) for vertical fall, where \(d\) is the distance fallen, \(a\) is the acceleration (9.81 m/s\(^2\)), and \(t\) is the time elapsed.
  • This aids in finding how long it takes to fall a given distance.
    This formula simplifies calculations by setting initial vertical velocity \(u\) to zero in free fall equations.
By understanding this model of uniformly accelerated motion, students can predict how objects behave under gravity when they start from rest or have an initial velocity.
This understanding interconnects with calculating both vertical and horizontal components of projectile motion.

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Most popular questions from this chapter

An object undergoing parabolic motion travels \(100 \mathrm{~m}\) in the horizontal direction before returning to its initial height. If the object is thrown initially at a \(30^{\circ}\) angle, determine the \(x\) component and the \(y\) component of the initial velocity. (Neglect any effects due to air resistance.)

Medical In a laboratory test of tolerance for high angular acceleration, pilots were swung in a circle \(13.4 \mathrm{~m}\) in diameter. It was found that they blacked out when they were spun at \(30.6 \mathrm{rpm}\) (rev/min). (a) At what acceleration (in SI units and in multiples of \(g\) ) did the pilots black out? (b) If you want to decrease the acceleration by \(25.0 \%\) without changing the diameter of the circle, by what percent must you change the time for the pilot to make one spin?

You drop a rock from rest from the top of a tall building. (a) How far has the rock fallen in \(2.50\) s? (b) What is the velocity of the rock after it has fallen \(11.0 \mathrm{~m}\) ? (c) It takes \(0.117 \mathrm{~s}\) for the rock to pass a \(2.00-\mathrm{m}\) high window. How far from the top of the building is the top of the window?

A car races at a constant speed of \(330 \mathrm{~km} / \mathrm{h}\) around a flat, circular track \(1.00 \mathrm{~km}\) in diameter. What is the car's radial acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) ?

A Chinook salmon can jump out of water with a speed of \(6.3 \mathrm{~m} / \mathrm{s}\). How far horizontally can a Chinook salmon travel through the air if it leaves the water with an initial angle \(40^{\circ}\) ? (Neglect any effects due to air resistance.)
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